A General Framework for Portfolio Theory. Part I: theory and various models
AA General Framework for Portfolio Theory. Part I:theory and various models
Stanislaus Maier-Paape (cid:3)
Qiji Jim Zhu y October 11, 2017
Abstract
Utility and risk are two often competing measurements on the investment suc-cess. We show that efficient trade-off between these two measurements for invest-ment portfolios happens, in general, on a convex curve in the two dimensional spaceof utility and risk. This is a rather general pattern. The modern portfolio theoryof Markowitz [15] and its natural generalization the capital market pricing model[22] are special cases of our general framework when the risk measure is taken tobe the standard deviation and the utility function is the identity mapping. Usingour general framework we also recover the results in [20] that extends the capitalmarket pricing model to allow for the use of more general deviation measures. Thisgeneralized capital asset pricing model also applies to e.g. when an approxima-tion of the maximum drawdown is considered as a risk measure. Furthermore, theconsideration of a general utility function allows to go beyond the \additive" per-formance measure to a \multiplicative" one of cumulative returns by using the logutility. As a result, the growth optimal portfolio theory [9] and the leverage spaceportfolio theory [28] can also be understood under our general framework. Thus,this general framework allows a uni(cid:12)cation of several important existing portfoliotheories and goes much beyond.
Key words.
Convex programming, (cid:12)nancial mathematics, risk measures, utility func-tions, efficient frontier, Markowitz portfolio theory, capital market pricing model, growthoptimal portfolio, fractional Kelly allocation.
AMS classi(cid:12)cation.
Acknowledgement.
We thank Andreas Platen for his constructive suggestions afterreading earlier versions of the manuscript. (cid:3)
Institut f(cid:127)ur Mathematik, RWTH Aachen University, 52062 Aachen, Templergraben 55, Germany [email protected] y Department of Mathematics, Western Michigan University, 1903 West Michigan Avenue, Kalamazoo,MI 49008, [email protected] Introduction
The Markowitz modern portfolio theory [15] pioneered the quantitative analysis of (cid:12)nan-cial economics. The most important idea proposed in this theory is that one should focuson the trade-off between expected return and the risk measured by the standard devi-ation. Mathematically, the modern portfolio theory leads to a quadratic optimizationproblem with linear constraints. Using this simple mathematical structure Markowitzgave a complete characterization of the efficient frontier for trade-off the return and risk.Tobin [26] showed that the efficient portfolios as an affine function of the expected return.Markowitz portfolio theory was later generalized by Lintner [9], Mossin [17], Sharpe [22]and Treynor [25] in the capital asset pricing model (CAPM) by involving a riskless bond.In the CAPM model, both the efficient frontier and the related efficient portfolios areaffine in terms of the expected return [22, 26].The nice structures of the solutions in the modern portfolio theory and the CAPMmodel afford many applications. For example, the CAPM model is designed to providereasonable price for risky assets in the market place. Sharpe used the ratio of excess returnto risk (called the Sharpe ratio) to provide a measurement for investment performance[23]. Also the affine structure of the efficient portfolio in terms of the expected returnleads to the concept of a market portfolio as well as the two fund theorem [26] and theone fund theorem [22, 26]. These results provided a theoretical foundation for passiveinvestment strategies.While using the expected return and standard deviation as measures for reward andrisk of a portfolio brings much convenience in the mathematical analysis, many othermeasures are more realistic. Since Bernoulli studied the St. Petersburg paradox [2],concave utility functions have been widely accepted as a more appropriate measure ofthe reward. General expected utilities have been used in many cases to measure theperformance of a portfolio. On the other hand, current drawdown [13], maximum draw-down and its approximations [10, 12, 30], deviation measure [20], conditional value atrisk [19] and more abstract coherent risk measures [1] are widely used as risk measures inpractices. A common thread in these risk measures is that they are convex re(cid:13)ecting thebelief that diversi(cid:12)cation reduces risk. The goal of this paper is to extend the modernportfolio theory into a general framework under which one can analyze efficient portfo-lios that trade-off between a convex risk measure and a reward captured by an expectedutility. We phrase our primal problem as a convex portfolio optimization problem ofminimizing a convex risk measure subject to the constraint that the expected utility ofthe portfolio is above a certain level. Thus, convex duality plays a crucial role and thestructure of the solutions to both the primal and dual problems often have signi(cid:12)cant(cid:12)nancial implications. We show that, in the space of risk measure and expected utility,efficient trade-off happens on an increasing concave curve. We also show that the efficientportfolios continuously depend on the level of the expected utility.The Markowitz modern portfolio theory and the capital asset pricing model are, ofcourse, special cases of this general theory. Markowitz determines portfolios of purely2isky assets which provide an efficient trade-off between expected return and risk mea-sured by the standard deviation (or equivalently the variance). Mathematically, this isa class of convex programming problems of minimizing the standard deviation of theportfolio parameterized by the level of the expected returns. The capital asset pricingmodel, in essence, extends the Markowitz modern portfolio theory by including a risklessbond in the portfolio. We observe that the space of the risk-expected return is, in fact,the space corresponding to the dual of the Markowitz portfolio problem. The shape ofthe famous Markowitz bullet is a manifestation of the well known fact that the optimalvalue function of a convex programming problem is convex with respect to the level ofconstraint. As mentioned above, the Markowitz portfolio problem is a quadratic opti-mization problem with linear constraint. This special structure of the problem dictatesthe affine structure of the optimal portfolio as a function of the expected return (see The-orem 4.1). This affine structure leads to the important two fund theorem that providesa theoretical foundation for the passive investment method. For the capital asset pricingmodel, such an affine structure appear in both the primal and dual representation of thesolutions which leads to the two fund separation theorem in the portfolio space and thecapital market line in the dual space of risk-return trade-off (cf. Theorem 4.5).The (cid:13)exibility in choosing different risk measures allows us to extend the analysisof the essentially quadratic risk measure pioneered by Markowitz to a wider range. Forexample, when the risk measure is a deviation measure [20], which happens e.g. whenan approximation of the current drawdown is considered (see [14]), and the expectedreturn is used to gauge the performance we show that the affine structure of the efficientsolution in the classical capital market pricing model is preserved (cf. Theorem 5.1),recovering in particular the results in [20]. This is signi(cid:12)cant in that it shows that thepassive investment strategy is justi(cid:12)able in a wide range of settings.The consideration of a general utility function, however, allows us to go beyond the\additive" performance measure in modern portfolio theory to a \multiplicative" oneincluding cumulative returns when, for example, using the log utility. As a result thegrowth optimal portfolio theory [9] and the leverage space portfolio theory [28] can alsobe understood under our general framework. The optimal growth portfolio pursues tomaximize the expected log utility which is equivalent to maximize the expected cumula-tive compound return. It is known that the growth optimal portfolio is usually too risky.Thus, practitioners often scale back the risky exposure from a growth optimal portfolio.In our general framework, we consider the portfolio that minimizes a risk measure givena (cid:12)xed level of expected log utility. Under reasonable conditions, we show that suchportfolios form a path parameterized by the level of expected log utility in the portfoliospace that connects the optimal growth portfolio and the portfolio of a riskless bond (seeTheorem 6.4). In general, for different risk measures we will derive different paths. Thesepaths provide justi(cid:12)cations for risk reducing curves proposed in the leverage space port-folio theory [28]. The dual problem projects the efficient trade-off path into a concavecurve in the risk-expected log utility space parallel to the role of Markowitz bullet in themodern portfolio theory and the capital market line in the capital asset pricing model.3nlike the modern portfolio theory and the capital asset pricing model, under the no ar-bitrage assumption, the efficient frontier here is usually a (cid:12)nite increasing concave curve.The lower left endpoint of the curve corresponds to the portfolio of pure riskless bond andthe upper right endpoint corresponds to the growth optimal portfolio. The increasingnature of the curve tells us that the more risk we take the more cumulative return wecan expect. The concavity of the curve indicates, however, that with the increase of therisk the marginal increase of the expected cumulative return will decrease. Thus, a riskaverse investor will usually not choose the optimal growth portfolio. It is also interest-ing to observe that considering the dual problem corresponding to the growth optimalportfolio problem will leads to a version of the fundamental theorem of asset pricing (seeTheorem 6.10) that connects the existence of an equivalent martingale measure to noarbitrage.Besides unifying the several important results laid out above, the general frameworkhas many new applications. In this (cid:12)rst installment of the paper, we layout the frame-work, derive the theoretical results of crucial importance and illustrate them with a fewexamples. More new applications will appear in the subsequent papers [3, 14]. We ar-range the paper as follows: First we discuss necessary preliminaries in the next section.Section 3 is devoted to our main result: a framework to trade-off between risk and utilityof portfolios and its properties. In Section 4 we give a uni(cid:12)ed treatment of Markowitzportfolio theory, capital asset pricing model, and the Sharpe ratio. Section 5 is devotedto a discussion on the conditions under which the optimal trade-off portfolio possessesan affine structure. Section 6 discusses growth optimal portfolio theory and leverageportfolio theory. We also highlight some related important applications such as the fun-damental theorem of asset pricing. We conclude in Section 7 pointing to applicationsworthy of further investigation.
We consider a simple one period (cid:12)nancial market model S on an economy with (cid:12)nitestates represented by a sample space Ω = f ! ; ! ; : : : ; ! N g . We use a probability space(Ω ; Ω ; P ) to represent the states of the economy and their corresponding probability ofoccurring, where 2 Ω is the algebra of all subsets of Ω. The space of random variables on(Ω ; Ω ; P ) is denoted RV (Ω ; Ω ; P ) and it is used to represent the payoff of risky (cid:12)nancialassets. Since the sample space Ω is (cid:12)nite, RV (Ω ; Ω ; P ) is a (cid:12)nite dimensional vectorspace. We use RV + (Ω ; Ω ; P ) to represent of the cone of nonnegative random variablesin RV (Ω ; Ω ; P ). Introducing the inner product ⟨ X; Y ⟩ = E [ XY ] ; X; Y RV (Ω ; Ω ; P ) ;RV (Ω ; Ω ; P ) becomes a ((cid:12)nite dimensional) Hilbert space.4 e(cid:12)nition 2.1. (Financial Market) We say that S t = ( S t ; S t ; : : : ; S Mt ) ; t = 0 ; is a(cid:12)nancial market in a one period economy provided that S R M +1+ and S (0 ; ) (cid:2) RV + (Ω ; Ω ; P ) M . Here S = 1 ; S = R > represents a risk free bond with a positivereturn when R > . The rest of the components S mt ; m = 1 ; : : : ; M represent the price ofthe m -th risky (cid:12)nancial asset at time t . We will use the notation b S t = ( S t ; (cid:1) (cid:1) (cid:1) ; S Mt ) when we need to focus on the riskyassets. We assume that S is a constant vector representing the prices of the assets inthis (cid:12)nancial market at t = 0. The risk is modeled by assuming b S = ( S ; : : : ; S M )to be a nonnegative random vector on the probability space (Ω ; Ω ; P ), that is S m RV + (Ω ; Ω ; P ) ; m = 1 ; ; : : : ; M . A portfolio is a column vector x R M +1 whose compo-nents x m represent the share of the m -th asset in the portfolio and S mt x m is the portionof capital invested in asset m at time t . Hence x corresponds to the investment in therisk free bond and b x = ( x ; : : : ; x M ) ⊤ is the risky part.We often need to restrict the selection of portfolios. For example, in many applicationswe consider only portfolios with unit initial cost, i.e. S (cid:1) x = 1. Thus, the followingde(cid:12)nition. De(cid:12)nition 2.2. (Admissible Portfolio)
We say that A (cid:26) R M +1 is a set of admissibleportfolios provided that A is a nonempty closed and convex set. We say that A is a setof admissible portfolios with unit initial price provided that A is a closed convex subsetof f x R M +1 : S (cid:1) x = 1 g . Let X be a (cid:12)nite dimensional Banach space. Recall that a set C (cid:26) X is convex if,for any x; y C and s [0 ; sx + (1 (cid:0) s ) y C . For an extended valued function f : X ! R [ f + we de(cid:12)ne its domain bydom( f ) := f x X : f ( x ) < and its epigraph by epi( f ) := f ( x; r ) X (cid:2) R : r (cid:21) f ( x ) g : We say f is lower semicontinuous if epi( f ) is a closed set. The following propositioncharacterizes an epigraph of a function. Proposition 2.3. (Characterization of Epigraph)
Let F be a closed subset of X (cid:2) R such that inf f r : ( x; r ) F g > (cid:0)1 for all x R . Then F is the epigraph for a lowersemicontinuous function f : X ! ( (cid:0)1 ; ] , i.e. F = epi( f ) , if and only if ( x; r ) F ) ( x; r + k ) F; k > : (2.1) Proof.
The key is to observe that, for a set F with the structure in (2.1), a function f ( x ) = inf f r : ( x; r ) F g (2.2)5s well de(cid:12)ned and then F = epi( f ) holds. Q.E.D.We say a function f is convex if epi( f ) is a convex set. Alternatively, f is convex ifand only if, for any x; y dom( f ) and s [0 ; f ( sx + (1 (cid:0) s ) y ) (cid:20) sf ( x ) + (1 (cid:0) s ) f ( y ) : Consider f : X ! [ (cid:0)1 ; + ). We say f is concave when (cid:0) f is convex and we say f isupper semicontinuous if (cid:0) f is lower semicontinuous. De(cid:12)ne the hypograph of a function f by hypo( f ) = f ( x; r ) X (cid:2) R : r (cid:20) f ( x ) g : Then a symmetric version of Proposition 2.3 is
Proposition 2.4. (Characterization of Hypograph)
Let F be a closed subset of X (cid:2) R such that sup f r : ( x; r ) F g < + for all x R . Then F is the hypograph of an uppersemicontinuous function f : X ! [ (cid:0)1 ; ) , i.e. F = hypo( f ) , if and only if ( x; r ) F ) ( x; r (cid:0) k ) F; k > : (2.3) Moreover, the function f can be de(cid:12)ned by f ( x ) = sup f r : ( x; r ) F g : (2.4) Remark 2.5.
The value of the function f in Proposition 2.3 (Proposition 2.4) at a givenpoint x is (cid:0)1 ( + ) if and only if f x g (cid:2) R (cid:26) F . Since utility functions are concave and risk measures are usually convex, the analysisof a general trade-off between utility and risk naturally leads to a convex programmingproblem. The general form of such convex programming problems is v ( y; z ) := inf x X [ f ( x ) : g ( x ) (cid:20) y; h ( x ) = z ] ; for y R M ; z R N ; (2.5)where f , g and h satisfy the following assumption. Assumption 2.6.
Assume that f : X ! R [ f + is a lower semicontinuous extendedvalued convex function, g : X ! R M is a vector valued function with convex components, (cid:20) signi(cid:12)es componentwise minorization and h : X ! R N is an affine mapping, fornatural numbers M; N . Moreover, at least one of the components of g has compactsublevel sets. Convex programming problems have nice properties due to the convex structure. Webrie(cid:13)y recall the pertinent results related to convex programming. First the optimalvalue function v is convex. This is a well-known result that can be found in standardbooks on convex analysis, e.g. [4]. It is, however, crucial for our applications below and,thus, we list it as a lemma and give a brief proof below for completeness.6 roposition 2.7. (Convexity of Optimal Value Function) Let f , g and h satisfy As-sumption 2.6. Then the optimal value function v in the convex programming problem(2.5) is convex and lower semicontinuous. Proof.
Consider ( y i ; z i ) dom( v ) ; i = 1 ; v and an arbitrary " > x i" feasible to the constraint of problem v ( y i ; z i ) such that f ( x i" ) < v ( y i ; z i ) + "; i = 1 ; : (2.6)Now for any (cid:21) [0 ; f ( (cid:21)x " + (1 (cid:0) (cid:21) ) x " ) (cid:20) (cid:21)f ( x " ) + (1 (cid:0) (cid:21) ) f ( x " ) (2.7) < (cid:21)v ( y ; z ) + (1 (cid:0) (cid:21) ) v ( y ; z ) + ": It is easy to check that (cid:21)x " + (1 (cid:0) (cid:21) ) x " is feasible for the problem v ( (cid:21) ( y ; z ) + (1 (cid:0) (cid:21) )( y ; z )). Thus, v ( (cid:21) ( y ; z ) + (1 (cid:0) (cid:21) )( y ; z )) (cid:20) f ( (cid:21)x " + (1 (cid:0) (cid:21) ) x " ). Combining withinequality (2.7) and letting " ! v ( (cid:21) ( y ; z ) + (1 (cid:0) (cid:21) )( y ; z )) (cid:20) (cid:21)v ( y ; z ) + (1 (cid:0) (cid:21) ) v ( y ; z ) ; that is to say v is convex.The lower semicontinuity of v is easier to verify. Q.E.D.By and large, there are two (equivalent) general approaches to help solving a convexprogramming problem: by using the related dual problem and by using Lagrange multi-pliers. The two methods are equivalent in the sense that a solution to the dual problemis exactly a Lagrange multiplier (see [5]). Using Lagrange multipliers is more accessibleto practitioners outside the special area of convex analysis. We will take this approach.The Lagrange multipliers method tells us that under mild assumptions we can expectthere exists a Lagrange multiplier (cid:21) = ( (cid:21) y ; (cid:21) z ) with (cid:21) y (cid:21) x is a solution tothe convex programming problem (2.5) if and only if it is a solution to the unconstrainedproblem of minimizing L ( x; (cid:21) ) := f ( x ) + ⟨ (cid:21); ( g ( x ) (cid:0) y; h ( x ) (cid:0) z ) ⟩ = f ( x ) + ⟨ (cid:21) y ; g ( x ) (cid:0) y ⟩ + ⟨ (cid:21) z ; h ( x ) (cid:0) z ⟩ : (2.8)The function L ( x; (cid:21) ) is called the Lagrangian. To understand why and when does aLagrange multiplier exist, we need to recall the de(cid:12)nition of the subdifferential. De(cid:12)nition 2.8. (Subdifferential)
Let X be a (cid:12)nite dimensional Banach space and X (cid:3) its dual space. The subdifferential of a lower semicontinuous convex function ϕ : X ! R [ f + at x dom( ϕ ) is de(cid:12)ned by @ϕ ( x ) = f x (cid:3) X (cid:3) : ϕ ( y ) (cid:0) ϕ ( x ) (cid:21) ⟨ x (cid:3) ; y (cid:0) x ⟩ 8 y X g : v . We summarize and prove the sufficiency in the lemma below which we will actuallyuse. Theorem 2.9. (Lagrange Multiplier)
Let v : R M (cid:2) R N ! R [ f + be the optimalvalue function of the constrained optimization problem (2.5) with f; g and h satisfyingAssumption 2.6. Suppose that, for (cid:12)xed ( y; z ) R M (cid:2) R N , (cid:0) (cid:21) = (cid:0) ( (cid:21) y ; (cid:21) z ) @v ( y; z ) and (cid:22) x is a solution of (2.5). Then(i) (cid:21) y (cid:21) ,(ii) the Lagrangian L ( x; (cid:21) ) de(cid:12)ned in (2.8) attains a global minimum at (cid:22) x , and(iii) (cid:21) satis(cid:12)es the complementary slackness condition ⟨ (cid:21); ( g ((cid:22) x ) (cid:0) y; h ((cid:22) x ) (cid:0) z ) ⟩ = ⟨ (cid:21) y ; g ((cid:22) x ) (cid:0) y ⟩ = 0 : (2.9) Proof.
Observe that v ( y; z ) is a nonincreasing function with respect to the minoriza-tion (cid:20) in y . Using (cid:0) (cid:21) @v ( y; z ), for any vector ∆ y (cid:21)
0, we have0 (cid:21) v ( y + ∆ y; z ) (cid:0) v ( y; z ) (cid:21) ⟨(cid:0) (cid:21); (∆ y; ⟩ : It follows that (cid:21) y (cid:21) v ( g ((cid:22) x ) ; h ((cid:22) x )) = v ( y; z ), wethen have 0 = v ( g ((cid:22) x ) ; h ((cid:22) x )) (cid:0) v ( y; z ) (cid:21) ⟨(cid:0) (cid:21); ( g ((cid:22) x ) (cid:0) y; h ((cid:22) x ) (cid:0) z ) ⟩ (cid:21) : It follows that the complementary slackness condition ⟨ (cid:21); ( g ((cid:22) x ) (cid:0) y; h ((cid:22) x ) (cid:0) z ) ⟩ = 0 (2.10)in (iii) holds.Finally, by the de(cid:12)nition of the subdifferential we have v ( g ( x ) ; h ( x )) (cid:0) v ( y; z ) (cid:21) ⟨(cid:0) (cid:21); ( g ( x ) (cid:0) y; h ( x ) (cid:0) z ) ⟩ : Thus, for any x , L ( x; (cid:21) ) = f ( x ) + ⟨ (cid:21); ( g ( x ) (cid:0) y; h ( x ) (cid:0) z ) ⟩ (2.11) (cid:21) v ( g ( x ) ; h ( x )) + ⟨ (cid:21); ( g ( x ) (cid:0) y; h ( x ) (cid:0) z ) ⟩(cid:21) v ( y; z )Using the fact that (cid:22) x is a solution to problem in (2.5) and the complementary slacknesscondition (2.10) we have v ( y; z ) = f ((cid:22) x ) = f ((cid:22) x ) + ⟨ (cid:21); ( g ((cid:22) x ) (cid:0) y; h ((cid:22) x ) (cid:0) z ) ⟩ = L ((cid:22) x; (cid:21) ) : (2.12)Combining (2.11) and (2.12) veri(cid:12)es (ii). Q.E.D.8 emark 2.10. By Theorem 2.9 Lagrange multipliers exist when (2.5) has a solution (cid:22) x and @v ( y; z ) ̸ = ∅ . Calculating @v ( y; z ) requires to know the value of v in a neighborhoodof ( y; z ) and is not realistic. Fortunately, the well-known Fenchel-Rockafellar theorem (seee.g. [4]) tells us when ( y; z ) belongs to the relative interior of dom( v ), then @v ( y; z ) ̸ = ∅ .This is a very useful sufficient condition. A particularly useful special case is the Slatercondition (see also [4]): when there is only an inequality constraint g ( x ) (cid:20) y , if thereexists x dom( f ) such that g ( x ) < y implies already that @v ( y ) ̸ = ∅ . We consider the (cid:12)nancial market described in De(cid:12)nition 2.1 and consider a set of admis-sible portfolios A (cid:26) R M +1 (see De(cid:12)nition 2.2). The payoff of each portfolio x A attime t = 1 is S (cid:1) x . The merit of a portfolio x is often judged by its expected utility E [ u ( S (cid:1) x )] where u is an increasing concave utility function. The increasing property of u models the more payoff the better. The concavity re(cid:13)ects the fact that with the increaseof payoff, its marginal utility to an investor decreases. On the other hand investors areoften sensitive to the risk of a portfolio which can be gauged by a risk measure. Becausediversi(cid:12)cation reduces risk, the risk measure should be a convex function. Some standard assumptions on the utility and risk functions are often needed in the moretechnical discussion below. We collect them here.
Assumption 3.1. (Conditions on Risk Measure)
Consider a continuous risk function r : A ! [0 ; + ) where A is a set of admissible portfolios according to De(cid:12)nition 2.2. Wewill often refer to some of the following assumptions. (r1) (Riskless Asset Contributes No risk) The risk measure r ( x ) = b r ( b x ) is a function ofonly the risky part of the portfolio, where x = ( x ; b x ) ⊤ . (r1n) (Normalization) There is at least one portfolio of purely bonds in A . Furthermore, r ( x ) = 0 if and only if x contains only riskless bonds, i.e. x = ( x ; b ⊤ for some x R . (r2) (Diversi(cid:12)cation Reduces Risk) The risk function r is convex. (r2s) (Diversi(cid:12)cation Strictly Reduces Risk) The risk function b r is strictly convex. (r3) (Positive homogeneous) For t > , b r ( t b x ) = t b r ( b x ) . Remark 3.2. (Deviation measure)
A risk measure satisfying assumptions (r1), (r1n),(r2) and (r3) is strongly related to a deviation measure in [20]. It is also related to thecoherent risk measure introduced in [1]. ssumption 3.3. (Conditions on Utility Function) Utility functions u : R ! R [ f(cid:0)1g are usually assumed to satisfy some of the following properties. (u1) (Pro(cid:12)t Seeking) The utility function u is an increasing function. (u2) (Diminishing Marginal Utility) The utility function u is concave. (u2s) (Strict Diminishing Marginal Utility) The utility function u is strictly concave. (u3) (Bankrupcy Forbidden) For t < , u ( t ) = (cid:0)1 . (u4) (Unlimited Growth) For t ! + , we have u ( t ) ! + . Another important condition which often appears in the (cid:12)nancial literature is noarbitrage.
De(cid:12)nition 3.4. (No Arbitrage)
We say a portfolio x R M +1 is an arbitrage on the(cid:12)nancial market S if ( S (cid:0) RS ) (cid:1) x (cid:21) S (cid:0) RS ) (cid:1) x ̸ = 0 : We say market S t has no arbitrage if there does not exist any arbitrage portfolio for the(cid:12)nancial market S t . An arbitrage is a way to make return above the risk free rate without taking any riskof losing money. If such an opportunity exists then investors will try to take advantageof it. In this process they will bid up the price of the risky assets and cause the arbitrageopportunity to disappear. For this reason, usually people assume a (cid:12)nancial market doesnot contain any arbitrage.The following is a weaker requirement than arbitrage:
De(cid:12)nition 3.5. (No Nontrivial Riskless Portfolio)
We say a portfolio x R M +1 is riskless if ( S (cid:0) RS ) (cid:1) x (cid:21) : We say the market has no nontrivial riskless portfolio if there does not exist a risklessportfolio x with b x ̸ = b . A trivial riskless portfolio of investing everything in the riskless asset S t always exists.A nontrivial riskless portfolio, however, is not to be expected and we will often use thisassumption.It turns out that the difference between no nontrivial riskless portfolio and no arbi-trage is exactly the following: De(cid:12)nition 3.6. (Nontrivial Bond Replicating Portfolio)
We say that x = ( x ; b x ) ⊤ is a nontrivial bond replicating portfolio if b x ̸ = b and ( S (cid:0) RS ) (cid:1) x = 0 : Proposition 3.7.
Consider (cid:12)nancial market S t of De(cid:12)nition 2.1. There is no nontrivialriskless portfolio in S t if and only if S t has no arbitrage portfolio and no nontrivial bondreplicating portfolio. Proof.
The conclusion follows directly from De(cid:12)nitions 3.4, 3.5 and 3.6. Q.E.D.
Corollary 3.8.
No nontrivial riskless portfolio implies no arbitrage portfolio.
Assuming the (cid:12)nancial market has no arbitrage then no nontrivial riskless portfoliois equivalent to no nontrivial bond replicating portfolio and has the following character-ization.
Theorem 3.9. (Characterization of no Nontrivial Bond Replicating Portfolio)
Assumingthe (cid:12)nancial market S t in De(cid:12)nition 2.1 has no arbitrage. Then the following assertionsare equivalent: (i) There is no nontrivial bond replicating portfolio. (ii)
For every nontrivial portfolio x with b x ̸ = b , there exists some ! Ω such that ( S ( ! ) (cid:0) RS ) (cid:1) x < : (3.1)(ii*) For every risky portfolio b x ̸ = b , there exists some ! Ω such that ( b S ( ! ) (cid:0) R b S ) (cid:1) b x < : (3.2)(iii) The matrix G := S ( ! ) (cid:0) RS S ( ! ) (cid:0) RS : : : S M ( ! ) (cid:0) RS M S ( ! ) (cid:0) RS S ( ! ) (cid:0) RS : : : S M ( ! ) (cid:0) RS M ... ... ... ... S ( ! N ) (cid:0) RS S ( ! N ) (cid:0) RS : : : S M ( ! N ) (cid:0) RS M R N (cid:2) M (3.3) has rank M , in particular N (cid:21) M . Proof.
We use a cyclic proof. (i) ! (ii): If (ii) fails then ( S (cid:0) RS ) (cid:1) x (cid:21) x . By (i) x must be an arbitrage, which is a contradiction. (ii) ! (ii*):obvious. (ii*) ! (iii): If (iii) is not true then G (cid:1) b x = 0 has a nontrivial solution which isa contradiction to (3.2). (iii) ! (i): Assume that there exists a portfolio x (cid:3) with b x (cid:3) ̸ = b S (cid:0) RS ) (cid:1) x (cid:3) = 0. This implies that ( b S (cid:0) R b S ) (cid:1) b x (cid:3) = 0so that G b x (cid:3) = 0 which contradicts (iii). Q.E.D.A rather useful corollary of Theorem 3.9 is that any of the conditions (i){(iii) of thattheorem ensures the covariance matrix of the risky assets to be positive de(cid:12)nite.11 orollary 3.10. (Positive De(cid:12)nite Covariance Matrix) Assume the (cid:12)nancial market S t in De(cid:12)nition 2.1 has no nontrivial riskless portfolio. Then the covariant matrix of therisky assets (cid:6) := E [( b S (cid:0) E ( b S )) ⊤ ( b S (cid:0) E ( b S ))] (3.4)= ( E [( S i (cid:0) E ( S i ))( S j (cid:0) E ( S j ))]) i;j =1 ;:::;M ; is positive de(cid:12)nite. Proof.
We note that under the assumption of the corollary, for any nontrivial riskyportfolio b x , b S (cid:1) b x cannot be a constant. Otherwise, ( b S (cid:0) R b S ) (cid:1) b x would be a constantwhich contradicts S t has no nontrivial riskless portfolio. It follows that for any nontrivialrisky portfolio b x , V ar ( b S (cid:1) b x ) = b x ⊤ (cid:6) b x > : Thus, (cid:6) is positive de(cid:12)nite. Q.E.D.
Remark 3.11.
Corollary 3.10 shows that the standard deviation as a risk measure sat-is(cid:12)es the properties (r1), (r1n), (r2) and (r3) in Assumption 3.1.
We note that to increase the utility one often has to take on more risk and as a result therisk increases. The converse is also true. For example, if one allocates all the capital tothe riskless bond then there will be no risk but the price to pay is that one has to forgoall the opportunities to get a high payoff on risky assets so as to reduce the expectedutility. Thus, the investment decision of selecting an appropriate portfolio becomes oneof trading-off between the portfolio’s expected return and risk. To understand such atrade-off we de(cid:12)ne, for a set of admissible portfolios A (cid:26) R M +1 in De(cid:12)nition 2.2, the set G ( r ; u ; A ) := f ( r; (cid:22) ) : x A s:t: r (cid:21) r ( x ) ; (cid:22) (cid:20) E [ u ( S (cid:1) x )] g (cid:26) R ; (3.5)on the two dimensional risk-expected utility space for a given risk measure r and utility u . Given a (cid:12)nancial market S t and a portfolio x , we often measure risk by observing S (cid:1) x . Corollary 3.12. (Induced Risk Measure) (a) Fixing a (cid:12)nancial market S t as in De(cid:12)-nition 2.1. Suppose that (cid:26) : RV (Ω ; Ω ; P ) ! [0 ; + ) is a lower semicontinuous, convexand positive homogeneous function. Moreover, assume that (cid:26) ( S (cid:1) x ) = (cid:26) ( b S (cid:1) b x ) . Then r : A ! [0 ; + ) , r ( x ) := (cid:26) ( S (cid:1) x ) is a lower semicontinuous risk measure satisfyingproperties (r1), (r2) and (r3) in Assumption 3.1.(b) If the (cid:12)nancial market S t has no nontrivial riskless portfolio and (cid:26) is strictlyconvex then for a set A of admissible portfolios with unit initial cost, b r : A ! [0 ; + ) satis(cid:12)es (r2s) in Assumption 3.1. roof. Since x ! S (cid:1) x is a linear mapping, the risk measure r inherits the propertiesof (cid:26) so that it satis(cid:12)es properties (r1), (r2) and (r3) in Assumption 3.1. One sufficientcondition for ^ r to preserve the strict convexity of (cid:26) is that the matrix G in (3.3) is offull rank since all portfolios have unit initial cost. It follows from Theorem 3.9 that thiscondition follows from no nontrivial riskless portfolio in the (cid:12)nancial market S t . Q.E.D. Remark 3.13.
The following are two sufficient conditions ensuring (cid:26) ( S (cid:1) x ) = (cid:26) ( b S (cid:1) b x ) that are easy to verify: (1) When (cid:26) is invariant under adding constants, i.e., (cid:26) ( X ) = (cid:26) ( X + c ) , for any X RV (Ω ; Ω ; P ) and c R . A useful example is when (cid:26) is the standard deviation. (2) When (cid:26) is restricted to a set of admissible portfolios A with unit initial cost. Inthis case we can see that b r ( b x ) := (cid:26) ( R + ( b S (cid:0) R b S ) (cid:1) b x ) = (cid:26) ( S (cid:1) x ) : (3.6)Similarly, we are interested in when the expected utility x E [ u ( S (cid:1) x )] of S (cid:1) x isstrictly concave in x . Below is a set of useful sufficient conditions. Lemma 3.14. (Strict Concavity of Expected Utility)
Assume that (a) the (cid:12)nancial market S t has no nontrivial riskless portfolio, (b) the utility function u satis(cid:12)es condition (u2s) in Assumption 3.3, and (c) A is a set of admissible portfolios with unit initial cost as in De(cid:12)nition 2.2.Then the expected utility E [ u ( S (cid:1) x )] as a function of the portfolio x is strictly concaveon A . Proof.
Since u is concave so is x E [ u ( S (cid:1) x )]. To prove that this function is strictlyconcave on A , consider two distinct portfolios x ; x A . By assumption (c), both x and x have unit initial cost and thus b x ̸ = b x . Assumption (a) and Proposition 3.7implies that for the matrix G de(cid:12)ned in (3.3), G b x ̸ = G b x . Thus, using again the factthat both x and x have unit initial cost, we have S (cid:1) x = R + ( b S (cid:0) R b S ) (cid:1) b x ̸ = R + ( b S (cid:0) R b S ) (cid:1) b x = S (cid:1) x : The strictly concavity of x ! E [ u ( S (cid:1) x )] now follows from the strict concavity of theutility function u as assumed in (b). Q.E.D.When r ( x ) = (cid:26) ( S (cid:1) x ) is induced by (cid:26) as in Corollary 3.12 we also use the notation G ( (cid:26); u; A ). Clearly, if A ′ (cid:26) A then G ( r ; u ; A ′ ) (cid:26) G ( r ; u ; A ). The following assumption willbe needed in concrete applications. Assumption 3.15. (Compact Level Sets)
Either (a) for each (cid:22) R , f x R M +1 : (cid:22) (cid:20) E [ u ( S (cid:1) x )] ; x A g is compact or (b) for each r R , f x R M +1 : r (cid:21) r ( x ) ; x A g iscompact. roposition 3.16. Assume that A is a set of admissible portfolios as in De(cid:12)nition 2.2.We claim: (a) Assume that the risk measure r satis(cid:12)es (r2) in Assumption 3.1 and theutility function u satis(cid:12)es (u2) in Assumption 3.3. Then set G ( r ; u ; A ) is convex and ( r; (cid:22) ) ( r ; u ; A ) implies that, for any k > , ( r + k; (cid:22) ) ( r ; u ; A ) and ( r; (cid:22) (cid:0) k ) ( r ; u ; A ) . (b) Assume furthermore that Assumption 3.15 holds. Then G ( r ; u ; A ) is closed. Proof. (a) The property ( r; (cid:22) ) ( r ; u ; A ) implies that, for any k >
0, ( r + k; (cid:22) ) ( r ; u ; A ) and ( r; (cid:22) (cid:0) k ) ( r ; u ; A ) follows directly from the de(cid:12)nition of G ( r ; u ; A ).Suppose that ( r ; (cid:22) ) ; ( r ; (cid:22) ) ( r ; u ; A ) and s [0 ; x ; x A such that r i (cid:21) r ( x i ) and (cid:22) i (cid:20) E [ u ( S (cid:1) x i )] ; i = 1 ; : Then convexity of r in x yields sr + (1 (cid:0) s ) r (cid:21) s r ( x ) + (1 (cid:0) s ) r ( x ) (cid:21) r ( sx + (1 (cid:0) s ) x ) ; and (u2) gives s(cid:22) + (1 (cid:0) s ) (cid:22) (cid:20) s E [ u ( S (cid:1) x )] + (1 (cid:0) s ) E [ u ( S (cid:1) x )] (cid:20) E [ u ( S (cid:1) ( sx + (1 (cid:0) s ) x ))] : Thus, s ( r ; (cid:22) ) + (1 (cid:0) s )( r ; (cid:22) ) ( r ; u ; A )so that G ( r ; u ; A ) is convex.(b) Suppose that ( r n ; (cid:22) n ) ! ( r; (cid:22) ), for a sequence in G ( r ; u ; A ). Then there exists asequence x n A such that r n (cid:21) r ( x n ) and (cid:22) n (cid:20) E [ u ( S (cid:1) x n )] : (3.7)By Assumption 3.15 a subsequence of x n (denoted again by x n ) converges to, say, (cid:22) x A .Taking limits in (3.7) we arrive at r (cid:21) r ((cid:22) x ) and (cid:22) (cid:20) E [ u ( S (cid:1) (cid:22) x )] : (3.8)Thus, ( r; (cid:22) ) ( r ; u ; A ) and hence G ( r ; u ; A ) is a closed set. Q.E.D.Now we can represent a portfolio x A (cid:26) R M +1 as a point ( r ( x ) ; E [ u ( S (cid:1) x )]) ( r ; u ; A ) in the two dimensional risk-expected utility space. Investors prefer portfolioswith lower risk if the expected utility is the same or with higher expected utility giventhe same level of risk. De(cid:12)nition 3.17. (Efficient Portfolio)
We say that a portfolio x A is Pareto efficient provided that there does not exist any portfolio x ′ A such that either r ( x ′ ) (cid:20) r ( x ) and E [ u ( S (cid:1) x ′ )] > E [ u ( S (cid:1) x )] or r ( x ′ ) < r ( x ) and E [ u ( S (cid:1) x ′ )] (cid:21) E [ u ( S (cid:1) x )] : e(cid:12)nition 3.18. (Efficient Frontier) We call the set of images of all efficient portfoliosin the two dimensional risk-expected utility space the efficient frontier and denote it by G eff ( r ; u ; A ) . The next theorem characterizes efficient portfolios in the risk-expected utility space.
Theorem 3.19. (Efficient Frontier)
Efficient portfolios represented in the two dimen-sional risk-expected utility space are all located in the (non vertical or horizontal) bound-ary of the set G ( r ; u ; A ) . Proof.
If a portfolio x represented in the risk-expected utility space as ( r; (cid:22) ) is noton the (non vertical or horizontal) boundary of the G ( r ; u ; A ), then for " small enoughwe have either ( r (cid:0) "; (cid:22) ) ( r ; u ; A ) or ( r; (cid:22) + " ) ( r ; u ; A ). This means x can beimproved. Q.E.D.The following relationship is straightforward but very useful. Theorem 3.20. (Efficient Frontier of Subsystem)
Consider admissible portfolios
A; B .If B (cid:26) A then G eff ( r ; u ; A ) \ G ( r ; u ; B ) (cid:26) G eff ( r ; u ; B ) . Proof.
The conclusion directly follows from G ( r ; u ; B ) (cid:26) G ( r ; u ; A ). Q.E.D. Remark 3.21. (Empty Efficient Frontier) If ( (cid:11); b A for all (cid:11) R and the increasingutility function u has no upper bound then for any risk measure r satisfying (r1) and(r1n) in Assumption 3.1, f g (cid:2) R (cid:26) G ( r ; u ; A ) . By Proposition 3.16 [0 ; + ) (cid:2) R (cid:26)G ( r ; u ; A ) which implies that G eff ( r ; u ; A ) = ∅ . Thus, practically meaningful G ( r ; u ; A ) always correspond to sets of admissible portfolios A such that the initial cost S (cid:1) x for all x A is limited. Moreover, if the initial cost has a range and riskless bonds are includedin the portfolio, then we will see a vertical line segment on the (cid:22) axis and the efficientportfolio corresponds to the upper bound of this vertical line segments. Thus, it sufficesto consider sets of portfolios A with unit initial cost. In view of Remark 3.21, in this section we will consider a set of admissible portfolios A with unit initial cost as in De(cid:12)nition 2.2. By Proposition 3.16 we can view the set G ( r ; u ; A ) as an epigraph on the expected utility-risk space or a hypograph on the risk-expected utility space. By Propositions 2.3 and 2.4, the set G ( r ; u ; A ) naturally de(cid:12)nestwo functions (cid:13) ( (cid:22) ) := inf f r : ( r; (cid:22) ) ( r ; u ; A ) g (3.9)= inf f r ( x ) : E [ u ( S (cid:1) x )] (cid:21) (cid:22); x A g ; and (cid:23) ( r ) := sup f (cid:22) : ( r; (cid:22) ) ( r ; u ; A ) g (3.10)= sup f E [ u ( S (cid:1) x )] : r ( x ) (cid:20) r; x A g : roposition 3.22. (Function Related to the Efficient Frontier) Assume that, the riskmeasure r satis(cid:12)es (r2) in Assumption 3.1 and the utility function u satis(cid:12)es (u2) inAssumption 3.3. Furthermore, assume that Assumption 3.15 holds for a set of admissibleportfolios A with unit initial cost. Then the functions (cid:22) (cid:13) ( (cid:22) ) and r (cid:23) ( r ) areincreasing lower semicontinuous convex and increasing upper semicontinuous concave,respectively. Proof.
The conclusion follows directly from Propositions 2.3 and 2.4 since G ( r ; u; A )is closed and convex according to Proposition 3.16.Alternatively, we can also directly apply Proposition 2.7 to the second representationin (3.9) and (3.10) to derive the convexity and concavity of (cid:13) and (cid:23) , respectively.The increasing property of (cid:13) and (cid:23) follows directly from the second representation in(3.9) and (3.10), respectively. Q.E.D.It also follows Corollary 3.23. (Representation of Efficient Frontier)
Assume that the risk measure r satis(cid:12)es condition (r2) in Assumption 3.1 and the utility function u satis(cid:12)es condition(u2) in Assumption 3.3. Then, for any set of admissible portfolios A with unit initialcost as de(cid:12)ned in De(cid:12)nition 2.2, Pareto efficient portfolios G eff ( r ; u ; A ) represented inthe expected utility-risk space are all located on the graph of (cid:13) or (cid:23) , i.e., G eff ( r ; u ; A ) = graph (cid:13) ( (cid:22) ) = graph (cid:23) ( r ) : We have seen that the efficient trade-off between risk and expected utility of a portfoliocan be represented as the graph of a lower semicontinuous convex function (cid:22) (cid:13) ( (cid:22) )that relates the level of expected return (cid:22) to a minimum risk. Alternatively, thesepoints in the expected utility-risk space can also be represented as the graph of an uppersemicontinuous concave function r (cid:23) ( r ) that relates the level of risk r to a maximumpossible utility. We now turn to analyze how the corresponding efficient portfolios behave.Ideally we would want that each point on the efficient trade-off frontier corresponds toexactly one portfolio. For this purpose we need additional assumptions on risk measuresand utility functions. Theorem 3.24. (Efficient Portfolio Path)
Assume that the (cid:12)nancial market S t de(cid:12)nedin De(cid:12)nition 2.1 has no nontrivial riskless portfolio and that A is a set of admissibleportfolios with unit initial cost as in De(cid:12)nition 2.2. We also assume Assumption 3.15holds. In addition, suppose that one of the following conditions holds: (c1) The risk measure r satis(cid:12)es conditions (r1) and (r2s) in Assumption 3.1 and theutility function satis(cid:12)es conditions (u1) and (u2) in Assumption 3.3. (c2) The risk measure r satis(cid:12)es conditions (r1) and (r2) in Assumption 3.1 and theutility function satis(cid:12)es conditions (u1) and (u2s) in Assumption 3.3. hen, in case there exists some x A with E [ u ( S (cid:1) x )] (cid:12)nite, we can de(cid:12)ne (cid:22) max := sup f E [ u ( S (cid:1) x )] ; x A g > (cid:0)1 ; (3.11) r min := inf f r ( x ) ; x A g 2 [0 ; + ) ; (3.12) (cid:22) min := lim r r min sup f E [ u ( S (cid:1) x )] : r ( x ) (cid:20) r; x A g ; (3.13) and r max := lim (cid:22) " (cid:22) max inf f r ( x ) : E [ u ( S (cid:1) x )] (cid:21) (cid:22); x A g (3.14) and claim the following: (a) For (cid:22) ( (cid:22) min ; (cid:22) max ) there exists exactly one portfolio x ( (cid:22) ) on the efficient frontier G eff ( r ; u; A ) which corresponds to ( (cid:13) ( (cid:22) ) ; (cid:22) ) . Moreover, the mapping (cid:22) ! x ( (cid:22) ) iscontinuous on ( (cid:22) min ; (cid:22) max ) . Furthermore, when (cid:22) max and/or (cid:22) min are/is attainedby some x A the above statement holds on the interval ( (cid:22) min ; (cid:22) max ] , [ (cid:22) min ; (cid:22) max ) or [ (cid:22) min ; (cid:22) max ] . (b) For r ( r min ; r max ) there exists exactly one portfolio y ( r ) on the efficient frontier G eff ( r ; u; A ) which corresponds to ( r; (cid:23) ( r )) . Moreover, the mapping r ! y ( r ) iscontinuous on ( r min ; r max ) . Furthermore, when r min is a minimum and/or r max are/is attained by some x A , the above statement holds on the interval [ r min ; r max ) , ( r min ; r max ] , or [ r min ; r max ] . (c) If in addition, r satis(cid:12)es (r1n) in Assumption 3.1 then r min = 0 , (cid:22) min = u ( R ) and x ( (cid:22) min ) = y ( r min ) = (1 ; b ⊤ (see Figure 2). Proof. (a) We focus on the case when condition (c1) is satis(cid:12)ed and will commenton the modi(cid:12)cations needed for the similar case when (c2) is satis(cid:12)ed.Consider (cid:22) ( (cid:22) min ; (cid:22) max ). Then we can (cid:12)nd a portfolio (cid:22) x A with E [ u ( S (cid:1) (cid:22) x )] (cid:21) (cid:22) .By (3.12) r min (cid:20) r ((cid:22) x ). Thus, the set A (cid:22) := f x : (cid:22) (cid:20) E [ u ( S (cid:1) x )] ; r ( x ) (cid:20) r ((cid:22) x ) ; x A g isnonempty. Moreover, Assumption 3.15 ensures that A (cid:22) is compact. It follows that thereexists at least one portfolio x ( (cid:22) ) such that r ( x ( (cid:22) )) = inf f r ( x ) : x A (cid:22) g = inf f r ( x ) : (cid:22) (cid:20) E [ u ( S (cid:1) x )] ; x A g : Clearly, x ( (cid:22) ) corresponds to the point ( (cid:13) ( (cid:22) ) ; (cid:22) ) on the efficient frontier G eff ( r ; u; A ).Next we show the portfolio x ( (cid:22) ) is unique. Suppose that portfolios x ̸ = x bothcorrespond to ( (cid:13) ( (cid:22) ) ; (cid:22) ) and belong to A . Then we must have r ( x ) = r ( x ) = (cid:13) ( (cid:22) ) and E [ u ( S (cid:1) x i )] (cid:21) (cid:22); x i A; i = 1 ;
2. Since A is convex, x (cid:3) = ( x + x ) = A . Conditions(r2s) and (u2) imply that E [ u ( S (cid:1) x (cid:3) )] (cid:21) (cid:22) and due to the strict convexity of b r and (r1), r ( x (cid:3) ) = b r ( b x (cid:3) ) < (cid:13) ( (cid:22) ), a contradiction. Thus, the mapping (cid:22) ! x ( (cid:22) ) is well de(cid:12)ned.17inally, we show the continuity of x ( (cid:22) ) by contradiction. Suppose this mapping isdiscontinuous at (cid:22) . Then, for a (cid:12)xed positive number " >
0, there exists a sequence (cid:22) n ! (cid:22) such that ∥ x ( (cid:22) n ) (cid:0) x ( (cid:22) ) ∥ (cid:21) " where E [ u ( S (cid:1) x ( (cid:22) n ))] (cid:21) (cid:22) n and r ( x ( (cid:22) n )) = b r ( b x ( (cid:22) n )) (cid:20) (cid:13) ( (cid:22) n ) : (3.15)By Assumption 3.15 we may assume without loss of generality that x ( (cid:22) n ) converges tosome portfolio x (cid:3) with ∥ x (cid:3) (cid:0) x ( (cid:22) ) ∥ (cid:21) " . Furthermore, by Proposition 3.22 (cid:13) ( (cid:22) ) isconvex and, thus, is continuous in its domain (see e.g. [18, Theorem 10.4]). Takinglimits in (3.15) yields E [ u ( S (cid:1) x (cid:3) )] (cid:21) (cid:22) and b r ( b x (cid:3) ) = (cid:13) ( (cid:22) ) : (3.16)But the uniqueness of the efficient portfolio (3.16) implies that x (cid:3) = x ( (cid:22) ), which isa contradiction. If (cid:22) min and/or (cid:22) max is (cid:12)nite and attained at some x A then withthe same arguments as above the unique continuous portfolio extends to the respectivebound of ( (cid:22) min ; (cid:22) max ).The proof for the case when condition (c2) holds is similar. The only difference is thatuniqueness of the efficient portfolio now follows from the strict concavity of the mapping x ! E [ u ( S (cid:1) x )] (by Lemma 3.14) and the convexity of r ( x ).(b) We know by de(cid:12)nition graph (cid:13) ( (cid:22) ) = graph (cid:23) ( r ). Moreover, (cid:13) ( (cid:22) ) is convex and,therefore, continuous on ( (cid:22) min ; (cid:22) max ). Finally, by assumption (u1), (cid:13) ( (cid:22) ) is a strictlyincreasing function on ( (cid:22) min ; (cid:22) max ). Thus (cid:13) ( (cid:22) ) is invertible on ( (cid:22) min ; (cid:22) max ). Clearly, theinverse of (cid:13) ( (cid:22) ) is (cid:23) ( r ) whose corresponding domain is ( r min ; r max ). The relationships r = (cid:13) ( (cid:22) ) and (cid:22) = (cid:23) ( r ) characterize the pair of inverse functions (cid:13) and (cid:23) . De(cid:12)ning y ( r ) = x ( (cid:23) ( r )) the conclusion of (b) follows.(c) Since A contains only portfolios of unit initial cost, (1 ; b ⊤ A when (r1n) issatis(cid:12)ed. Then we can directly verify the conclusion in (c). Q.E.D. Remark 3.25. (a) When Assumption 3.15 (b) holds, then r min = min f r ( x ) : x A g and (cid:22) min = sup f E [ u ( S (cid:1) x )] : r ( x ) = r min ; x A g is also (cid:12)nite by (3.13). A typical efficient frontier corresponding to this case is illustratedin Figure 1.(b) It is possible that (cid:22) max and/or r max to be + . Suppose (cid:22) max is (cid:12)nite and attainedat an efficient portfolio x ( (cid:22) max ) . Under the conditions of the theorem the portfolio (cid:20) := x ( (cid:22) max ) is unique and independent of the risk measure. A graphic illustration is given inFigure 3.(c) Trade-off between utility and risk is thus implemented by portfolios x ( (cid:22) ) whichtrace out a curve in the leverage space of Vince [28]. Note that the curve x ( (cid:22) ) dependson the risk measure r as well as the utility function u . This provides a method forsystematically selecting portfolios in the leverage space to reduce risk exposure. (cid:22) G ( r ; u; A ) G eff ( r ; u; A ) Figure 1: Efficient frontier with both r min and (cid:22) min are (cid:12)nite and attained. r(cid:22) G ( r ; u; A ) G eff ( r ; u; A ) Figure 2: Efficient frontier with (1 ; b ⊤ A .19 (cid:22) G ( r ; u; A ) G eff ( r ; u; A ) Figure 3: Efficient frontier when r min > (cid:22) max is (cid:12)nite and attained as maximum. Let us now turn to applications of the general theory. We show that the results in theprevious section provide a general uni(cid:12)ed framework for several familiar portfolio theories.They are Markowitz portfolio theory, CAPM model, growth optimal portfolio theory andleverage space portfolio theory. Of course, when dealing with concrete risk measuresand expected utilities related to these concrete theories additional helpful structure inthe solutions often emerge. Although many different expositions of these theories doalready exist in the literature, for convenience of readers we include brief argumentsusing Lagrange multiplier methods. In this entire section we will assume that the market S t from De(cid:12)nition 2.1 has no nontrivial riskless portfolio. Markowitz [15] portfolio theory which considers only risky assets can be understood asa special case of the framework discussed in Section 3. The risk measure is the standarddeviation (cid:27) and the utility function is the identity function. So we face the problemmin (cid:27) ( b S (cid:1) b x ) (4.1)Subject to E [ b S (cid:1) b x ] (cid:21) (cid:22); b S (cid:1) b x = 1 : We assume E [ b S ] is not proportional to b S , that is, for any (cid:11) R , E [ b S ] ̸ = (cid:11) b S : (4.2)Since the variance is a monotone increasing function of the standard deviation we canminimize half of variance for convenience. 20in b x R M b r ( b x ) := 12 Var( b S (cid:1) b x ) = 12 (cid:27) ( b S (cid:1) b x ) = 12 b x ⊤ (cid:6) b x (4.3)Subject to E [ b S (cid:1) b x ] (cid:21) (cid:22); b S (cid:1) b x = 1 : Optimization problem (4.3) is already in the form (3.9) with A = f x R M +1 : S (cid:1) x =1 ; x = 0 g . We can check condition (c1) in Theorem 3.24 is satis(cid:12)ed. Moreover, Corollary3.10 implies that (cid:6) is positive de(cid:12)nite since S t has no nontrivial riskless portfolio. Hence,the risk function b r has compact level sets. Thus, Assumption 3.15 is satis(cid:12)ed and Theorem3.24 is applicable. Let b x ( (cid:22) ) be the optimal portfolio corresponding to (cid:22) . Consider theLagrangian L ( b x; (cid:21) ) := 12 b x ⊤ (cid:6) b x + (cid:21) ( (cid:22) (cid:0) E [ b S ] (cid:1) b x ) + (cid:21) (1 (cid:0) b S (cid:1) b x ) ; (4.4)where (cid:21) (cid:21)
0. Thanks for Theorem 2.9 we have0 = ∇ b x L = b x ⊤ ( (cid:22) )(cid:6) (cid:0) ( (cid:21) E [ b S ] + (cid:21) b S ) : (4.5)In other words b x ⊤ ( (cid:22) ) = ( (cid:21) E [ b S ] + (cid:21) b S )(cid:6) (cid:0) : (4.6)We must have (cid:21) > b x ⊤ ( (cid:22) ) would be unrelated to the payoff b S . Thecomplementary slackness condition implies that E [ b S (cid:1) b x ( (cid:22) )] = (cid:22) . Right multiplying (4.5)by b x ( (cid:22) ) we have (cid:27) ( (cid:22) ) = (cid:21) (cid:22) + (cid:21) : (4.7)To determine the Lagrange multipliers, we need the numbers (cid:11) = E [ b S ](cid:6) (cid:0) E [ b S ] ⊤ , (cid:12) = E [ b S ](cid:6) (cid:0) b S ⊤ and (cid:13) = b S (cid:6) (cid:0) b S ⊤ . Right multiplying (4.6) by E [ b S ] ⊤ and b S ⊤ we have (cid:22) = (cid:21) (cid:11) + (cid:21) (cid:12) (4.8)and 1 = (cid:21) (cid:12) + (cid:21) (cid:13): (4.9)Solving (4.8) and (4.9) we derive (cid:21) = (cid:13)(cid:22) (cid:0) (cid:12)(cid:11)(cid:13) (cid:0) (cid:12) and (cid:21) = (cid:11) (cid:0) (cid:12)(cid:22)(cid:11)(cid:13) (cid:0) (cid:12) ; (4.10)21 (cid:22)(cid:12)=(cid:13) p (cid:13) Figure 4: Markowitz Bulletwhere (cid:11)(cid:13) (cid:0) (cid:12) = det ([ E [ b S ] b S ] (cid:6) (cid:0) [ E [ b S ⊤ ] ; b S ⊤ ] ) > (cid:0) is positive de(cid:12)nite and condition (4.2) holds. Substituting (4.10) into (4.7) wesee that the efficient frontier is determined by the curve (cid:27) ( (cid:22) ) = √ (cid:13)(cid:22) (cid:0) (cid:12)(cid:22) + (cid:11)(cid:11)(cid:13) (cid:0) (cid:12) = √ (cid:13)(cid:11)(cid:13) (cid:0) (cid:12) ( (cid:22) (cid:0) (cid:12)(cid:13) ) + 1 (cid:13) (cid:21) p (cid:13) (4.12)usually referred to as the Markowitz bullet due to its shape. A typical Markowitz bulletis shown in Figure 4 with an asymptote (cid:22) = (cid:12)(cid:13) + (cid:27) ( (cid:22) ) √ (cid:11)(cid:13) (cid:0) (cid:12) (cid:13) : (4.13)Note that G ( Var ; id; f S (cid:1) x = 1 ; x = 0 g ) = G ( (cid:27); id; f S (cid:1) x = 1 ; x = 0 g ). Thus,relationships (4.12) and (4.13) describe the efficient frontier G eff ( (cid:27); id; f S (cid:1) x = 1 ; x = 0 g )as in De(cid:12)nition 3.18. Also note that (4.12) implies that (cid:22) min = (cid:12)=(cid:13) and r min = 1 = p (cid:13) .Thus, as a corollary of Theorem 3.24, we have Theorem 4.1. (Markowitz Portfolio Theorem)
Assume that the (cid:12)nancial market S t has no nontrivial riskless portfolio and E [ b S ] is not proportional to b S (see (4.2)). TheMarkowitz efficient portfolios of (4.1) represented in the ( (cid:27); (cid:22) ) (cid:0) plane are given by G eff ( (cid:27); id ; f S (cid:1) x = 1 ; x = 0 g ) : hey correspond to the upper boundary of the Markowitz bullet given by (cid:27) ( (cid:22) ) = √ (cid:13)(cid:22) (cid:0) (cid:12)(cid:22) + (cid:11)(cid:11)(cid:13) (cid:0) (cid:12) ; (cid:22) [ (cid:12)(cid:13) ; + ) : The optimal portfolio b x ( (cid:22) ) can be determined by (4.6) and (4.10) as b x ( (cid:22) ) = (cid:22) (cid:6) (cid:0) ( (cid:13) E [ b S ⊤ ] (cid:0) (cid:12) b S ⊤ ) (cid:11)(cid:13) (cid:0) (cid:12) + (cid:6) (cid:0) ( (cid:11) b S ⊤ (cid:0) (cid:12) E [ b S ⊤ ]) (cid:11)(cid:13) (cid:0) (cid:12) ; (4.14) which is affine in (cid:22) . The structure of the optimal portfolio in (4.14) implies the well known two fundtheorem derived by Tobin in [26].
Theorem 4.2. (Two Fund Theorem)
Select two distinct portfolios on the Markowitz ef-(cid:12)cient frontier. Then any portfolio on the Markowitz efficient frontier can be representedas the linear combination of these two portfolios.
Remark 4.3.
The two fund theorem can be viewed as the theoretical foundation forthe passive investment strategy of buy and hold broad based indices. Since most mutualfunds and hedge funds underperform the broad based indices, empirically we can regardbroad based indices such as SP500 and NASDAQ as Markowitz efficient portfolios. Bythe two fund theorem holding two such broad based indices passively we can produceany efficient portfolio on the Markowitz bullet.
The capital asset pricing model (CAPM) is a theoretical model independently proposedby Lintner [9], Mossin [17], Sharpe [22] and Treynor [25] for pricing a risky asset accordingto its expected payoff and market risk, often referred to as the beta. The core of thecapital asset pricing model is an extension of the Markowitz portfolio theory to includea riskless bond. Thus we can apply the general framework in Section 3 with the samesetting as in Section 4.1. Similar to the previous section we can consider the equivalentproblem of min x R M +1 (cid:27) ( S (cid:1) x ) = 12 b x ⊤ (cid:6) b x =: b r ( b x ) (4.15)Subject to E [ S (cid:1) x ] (cid:21) (cid:22);S (cid:1) x = 1 : Similar to the last section problem (4.15) is in the form (3.9) with A = f x R M +1 : S (cid:1) x = 1 g . We can check condition (c1) in Theorem 3.24 is satis(cid:12)ed. Again the risk23unction b r has compact level sets since (cid:6) is positive de(cid:12)nite. Thus, Assumption 3.15 issatis(cid:12)ed and Theorem 3.24 is applicable. The Lagrangian of this convex programmingproblem is L ( x; (cid:21) ) := 12 b x ⊤ (cid:6) b x + (cid:21) ( (cid:22) (cid:0) E [ S ] (cid:1) x ) + (cid:21) (1 (cid:0) S (cid:1) x ) ; (4.16)where (cid:21) (cid:21)
0. Again we have0 = ∇ x L = (0 ; b x ⊤ ( (cid:22) )(cid:6)) (cid:0) ( (cid:21) E [ S ] + (cid:21) S ) : (4.17)Using S = R and S = 1, the (cid:12)rst component of (4.17) implies (cid:21) = (cid:0) (cid:21) R: (4.18)So that (4.17) becomes0 = ∇ x L = (0 ; b x ⊤ ( (cid:22) )(cid:6)) (cid:0) (cid:21) ( E [ S ] (cid:0) RS ) : (4.19)Clearly (cid:21) > b x ( (cid:22) ) ̸ = 0. Using the complementary slackness condition E [ S (cid:1) x ( (cid:22) )] = (cid:22) we derive (cid:27) ( (cid:22) ) = b x ⊤ ( (cid:22) )(cid:6) b x ( (cid:22) ) = (cid:21) ( (cid:22) (cid:0) R ) ; (4.20)by right multiplying x ( (cid:22) ) in (4.19). Solving b x ⊤ ( (cid:22) ) from (4.19) we have b x ⊤ ( (cid:22) ) = (cid:21) ( E [ b S ] (cid:0) R b S )(cid:6) (cid:0) : (4.21)Right multiplying with E [ b S ⊤ ] and b S ⊤ and using the (cid:11); (cid:12) and (cid:13) introduced in the previoussection we derive (cid:22) (cid:0) x ( (cid:22) ) R = (cid:21) ( (cid:11) (cid:0) R(cid:12) ) (4.22)and 1 (cid:0) x ( (cid:22) ) = (cid:21) ( (cid:12) (cid:0) R(cid:13) ) ; (4.23)respectively. Multiplying (4.23) by R and subtract it from (4.22) we get (cid:22) (cid:0) R = (cid:21) ( (cid:11) (cid:0) (cid:12)R + (cid:13)R ) : (4.24)Combining (4.20) and (4.24) we arrive at (cid:27) ( (cid:22) ) = ( (cid:22) (cid:0) R ) (cid:11) (cid:0) (cid:12)R + (cid:13)R : (4.25)24t only makes sense to involve risky assets when we can expect an excess return. Thus, (cid:22) (cid:21) R . Relation (4.25) de(cid:12)nes a straight line on the ( (cid:27); (cid:22) )-plane (cid:27) ( (cid:22) ) = (cid:22) (cid:0) R p ∆ or (cid:22) = R + (cid:27) ( (cid:22) ) p ∆ ; (4.26)where ∆ := (cid:11) (cid:0) (cid:12)R + (cid:13)R > E [ b S ] (cid:0) R b S ̸ = 0 (4.27)since (cid:6) is positive de(cid:12)nite. The line given in (4.26) is called the capital market line .Also combining (4.21), (4.23) and (4.24) we have x ⊤ ( (cid:22) ) = ∆ (cid:0) [ (cid:11) (cid:0) (cid:12)R (cid:0) (cid:22) ( (cid:12) (cid:0) (cid:13)R ) ; ( (cid:22) (cid:0) R )( E [ b S ] (cid:0) R b S )(cid:6) (cid:0) ] : (4.28)Again we see the affine structure of the solution. In particular, when (cid:22) = R and (cid:22) =( (cid:11) (cid:0) (cid:12)R ) = ( (cid:12) (cid:0) (cid:13)R ) we derive, respectively, the portfolio (1 ; b ⊤ that contains only theriskless bond and the portfolio (0 ; ( E [ b S ] (cid:0) R b S )(cid:6) (cid:0) = ( (cid:12) (cid:0) (cid:13)R )) ⊤ that contains only riskyassets. We call this portfolio the market portfolio and denote it x M . The market portfoliocorresponds to the coordinates( (cid:27) M ; (cid:22) M ) = ( p ∆ (cid:12) (cid:0) (cid:13)R ; R + ∆ (cid:12) (cid:0) (cid:13)R ) : (4.29)Since the risk (cid:27) is non negative we see that the market portfolio exists only when (cid:12) (cid:0) (cid:13)R > : This condition is ( E [ b S ] (cid:0) R b S ) (cid:1) (cid:6) (cid:0) b S ⊤ > : (4.30)Note that (4.30) also implies (4.27).Again note that although the computation is done in terms of the risk function b r ( b x ) = b x ⊤ (cid:6) b x , relationships in (4.26) are in terms the risk function (cid:27) ( S (cid:1) x ). Thus, they describethe efficient frontier G eff ( (cid:27); id; S (cid:1) x = 1) as in De(cid:12)nition 3.18. In summary, we have Theorem 4.4. (CAPM)
Assume that the (cid:12)nancial market S t of De(cid:12)nition 2.1 has nonontrivial riskless portfolio. Moreover assume that condition (4.30) holds. The efficientportfolios for the CAPM model G eff ( (cid:27); id ; f S (cid:1) x = 1 g ) represented in the ( (cid:27); (cid:22) ) (cid:0) plane area straight line passing through (0 ; R ) corresponding to the portfolio of pure risk free bondand ( (cid:27) M ; (cid:22) M ) corresponding to the market portfolio of purely risky assets. The optimalportfolio x ( (cid:22) ) can be determined by (4.28) which is affine in (cid:22) . (cid:22) ( (cid:27) M ; (cid:22) M )(0 ; R ) Figure 5: Capital Market Line and Markowitz BulletBy Theorem 3.20( (cid:27) M ; (cid:22) M ) eff ( (cid:27); id ; f S (cid:1) x = 1 g ) \ G ( (cid:27); id ; f S (cid:1) x = 1 ; x = 0 g ) (4.31) (cid:26) G eff ( (cid:27); id ; f S (cid:1) x = 1 ; x = 0 g ) : Thus, the market portfolio has to reside on the Markowitz efficient frontier. Moreover,by (4.28) we can see that the market portfolio x M is the only portfolio on the CAPMefficient frontier that consists of purely risky assets. Thus, G eff ( (cid:27); id ; f S (cid:1) x = 1 g ) \ G ( (cid:27); id ; f S (cid:1) x = 1 ; x = 0 g ) = f ( (cid:27) M ; (cid:22) M ) g ; (4.32)so that the capital market line is tangent to the Markowitz bullet at ( (cid:27) M ; (cid:22) M ) as illus-trated in Figure 5. The affine structure of the solutions is summarized in the followingone fund theorem [22, 26]. Theorem 4.5. (One Fund Theorem)
Assume that the (cid:12)nancial market S t has no non-trivial riskless portfolio. Moreover assume that condition (4.30) holds. All the optimalportfolios in the CAPM model (4.15) are generalized convex combinations of the risklessbond and the market portfolio x M = (0 ; ( E [ b S ] (cid:0) R b S )(cid:6) (cid:0) = ( (cid:12) (cid:0) (cid:13)R )) ⊤ . Optimal portfolios x ( (cid:22) ) are affine in (cid:22) (see (4.28)) and can be represented as points in the ( (cid:27); (cid:22) ) -plane aslocated on the capital market line (cid:22) = R + (cid:27) p ∆ ; (cid:27) (cid:21) : The capital market line is tangent to the boundary of the Markowitz bullet at the co-ordinates of the market portfolio ( (cid:27) M ; (cid:22) M ) and intercepts the (cid:22) axis at (0 ; R ) (see Fig.5). Remark 4.6.
The one fund theorem combined with the two fund theorem provides atheoretical foundation for the passive investment strategy. The two fund theorem implies hat if two broad based indices are approximately on the Markowitz frontier then we canuse a linear combination of these two indices to derive the market portfolio. Thus, by theone fund theorem in order to construct an efficient portfolio in the sense of the CAPMmodel we only need to consider a mix of the bond and the two indices. Alternatively we can write the slope of the capital market line as p ∆ = (cid:22) M (cid:0) R(cid:27) M : (4.33)This quantity is called the price of risk and we can rewrite the equation for the capitalmarket line (4.26) as (cid:22) = R + (cid:22) M (cid:0) R(cid:27) M (cid:27): (4.34) Remark 4.7. (Sharpe Ratio)
We note that for any given portfolio x its correspondingpair of coordinates ( (cid:27); (cid:22) ) in the risk-return space also produces a ratio (cid:22) (cid:0) R(cid:27) : (4.35)
In the risk-return space this is the slope of the line representing portfolios mixing x witha riskless bond. Clearly the larger this ratio the better the portfolio serves this purpose.Sharpe [23] proposed to use this ratio, later called Sharpe ratio, to measure the perfor-mance of mutual funds. We can also use the capital market line to price a risky asset as we initially set out todo. The pricing principle in the capital asset pricing model is that adding a fair pricedrisky asset to the market should not change the capital market line. For convenience weassume that the price is implied by the expected return of the asset. Thus, given a riskyasset a i , we try to determine its expected return (cid:22) i . Theorem 4.8. (Capital Asset Pricing Model: the beta)
Assume that the (cid:12)nancial market S t of De(cid:12)nition 2.1 has no nontrivial riskless portfolio. Moreover assume that condition(4.30) holds. Let a i be the fair price of a risky asset a i with a payoff a i at t = 1 . Denotethe expected percentage return of a i by (cid:22) i = E [ a i ] =a i . Then (cid:22) i = R + (cid:12) i ( (cid:22) M (cid:0) R ) : (4.36) Here (cid:12) i := (cid:27) iM =(cid:27) M is called the beta of a i , where (cid:27) iM := Cov( a i =a i ; S (cid:1) x M ) is thecovariance of a i =a i and the payoff of the market portfolio. Proof.
Consider a portfolio relies on the parameter (cid:11) that mixes the risky asset a i and the market portfolio: p ( (cid:11) ) = (cid:11)a i =a i + (1 (cid:0) (cid:11) ) S (cid:1) x M : (4.37)27enote the expected return and the standard deviation of p ( (cid:11) ) by (cid:22) (cid:11) and (cid:27) (cid:11) , respectively.Hence we have (cid:22) (cid:11) = (cid:11)(cid:22) i + (1 (cid:0) (cid:11) ) (cid:22) M ; (4.38)and (cid:27) (cid:11) = (cid:11) (cid:27) i + 2 (cid:11) (1 (cid:0) (cid:11) ) (cid:27) iM + (1 (cid:0) (cid:11) ) (cid:27) M ; (4.39)where (cid:27) i is the variance of a i =a i . The parametric curve ( (cid:27) (cid:11) ; (cid:22) (cid:11) ) must lie below thecapital market line because the latter consists of optimal portfolios. On the other handit is clear that when (cid:11) = 0 this curve coincides with the capital market line. Thus, thecapital market line is tangent to the line of the parametric curve ( (cid:27) (cid:11) ; (cid:22) (cid:11) ) at (cid:11) = 0. Sincethe slope of the capital market line is ( (cid:22) M (cid:0) R ) =(cid:27) M , it follows that (cid:22) M (cid:0) R(cid:27) M = [ d(cid:22) (cid:11) d(cid:27) (cid:11) ] (cid:11) =0 = (cid:27) M ( (cid:22) i (cid:0) (cid:22) M ) (cid:27) iM (cid:0) (cid:27) M : (4.40)Solving for (cid:22) i we derive (cid:22) i = R + (cid:12) i ( (cid:22) M (cid:0) R ) : (4.41)Q.E.D. The affine dependence of the efficient portfolio on the return (cid:22) observed in the CAPM stillholds when the standard deviation is replaced by the more general deviation measure (see[20]. In this section we derive this affine structure using the general framework discussedin Section 3 and provide a proof different from that of [20]. We also construct a counter-example showing that the two fund theorem (Theorem 4.2) fails in this setting. Let’sconsider a risk measure r that satis(cid:12)es (r1), (r1n), (r2) and (r3) in Assumption 3.1 andthe related problem of (cid:12)nding efficient portfolios becomesmin x R M +1 r ( x ) = b r ( b x ) (5.1)Subject to E [ S (cid:1) x ] (cid:21) (cid:22);S (cid:1) x = 1 : Since for (cid:22) = R there is an obvious solution x ( R ) = (1 ; b
0) corresponding to r ( x ( R )) = b r ( b
0) = 0, we have r min = 0 and (cid:22) min = R . In what follows we will only consider (cid:22) > R . Moreover, we note that for b r satisfying the positive homogeneous property (r3)in Assumption 3.1, b y @ b r ( b x ) implies that b r ( b x ) = ⟨ b y; b x ⟩ : (5.2)28n fact, for any t ( (cid:0) ; t b r ( b x ) = b r ((1 + t ) b x ) (cid:0) b r ( b x ) (cid:21) t ⟨ b y; b x ⟩ ; (5.3)and (5.2) follows. Now we can state and prove the theorem on affine dependence of theefficient portfolio on the return (cid:22) . Theorem 5.1. (Affine Efficient Frontier for Positive Homogeneous Risk Measures)
As-sume that the (cid:12)nancial market S t of De(cid:12)nition 2.1 has no nontrivial riskless portfolio.Assume that the risk measure r satis(cid:12)es assumptions (r1), (r1n), (r2) and (r3) in As-sumption 3.1 with A = f x R M +1 : S (cid:1) x = 1 g and Assumption 3.15 (b) holds.Furthermore, assume that there exists some (cid:22) m ; ; : : : ; M g with E [ S (cid:22) m ] ̸ = RS (cid:22) m : (5.4) Then there exists an efficient portfolio x corresponding to ( r ; (cid:22) ) = ( r ( x ) ; R + 1) onthe efficient frontier for problem (5.1) such that the efficient frontier for problem (5.1)in the risk-expected return space is a straight line that passes through the points (0,R)corresponding to a portfolio of pure bond (1 ; b ⊤ and ( r ; (cid:22) ) corresponding to the portfolio x , respectively. Moreover, the straight line connecting (1 ; b ⊤ and x in the portfoliospace, namely for (cid:22) (cid:21) R , ( (cid:22) (cid:0) (cid:22) )(1 ; b ⊤ + ( (cid:22) (cid:0) R ) x (5.5) represents a set of efficient portfolios that corresponds to ( (cid:13) ( (cid:22) ) ; (cid:22) ) = (( (cid:22) (cid:0) R ) r ; (cid:22) ) (5.6) in the risk-expected return space (see De(cid:12)nition 3.18 and (3.9)). Proof.
The Lagrangian of this convex programming problem (5.1) is L ( x; (cid:21) ) := r ( x ) + (cid:21) ( (cid:22) (cid:0) E [ S ] (cid:1) x ) + (cid:21) (1 (cid:0) S (cid:1) x ) ; (5.7)where (cid:21) (cid:21) (cid:21) R .Condition (5.4) implies that, for any (cid:22) there exists a portfolio of the form y =( y ; ; : : : ; ; y (cid:22) m ; ; : : : ; ⊤ satisfying [ E [ S (cid:1) y ] S (cid:1) y ] = [ Ry + E [ S (cid:22) m ] y (cid:22) m y + S (cid:22) m y (cid:22) m ] = [ R E [ S (cid:22) m ]1 S (cid:22) m ] [ y y (cid:22) m ] = [ (cid:22) ] ; (5.8)because the matrix in (5.8) is invertible. Thus, for any (cid:22) (cid:21) R , Assumption 3.15 (b) with A = f x R M +1 : S (cid:1) x = 1 g and condition (5.4) ensure the existence of an optimalsolution to problem (5.1).Denoting one of those solutions by x ( (cid:22) ) (may not be unique) we have (cid:13) ( (cid:22) ) = r ( x ( (cid:22) )) = b r ( b x ( (cid:22) )) : (5.9)29ixing (cid:22) = R + 1 > R , denote x = x ( (cid:22) ). Then (cid:21) E [ S ] + (cid:21) S @ r ( x ) : (5.10)Since r is independent of x we have (cid:21) E [ S ] + (cid:21) S = 0 or (cid:21) = (cid:0) (cid:21) R: (5.11)Substituting (5.11) into (5.10) we have (cid:21) E [ b S (cid:0) R b S ] @ b r ( b x ) (5.12)so that, for all b x R M , b r ( b x ) (cid:0) b r ( b x ) (cid:21) (cid:21) E [( b S (cid:0) R b S ) (cid:1) ( b x (cid:0) b x )] = (cid:21) ( E [( b S (cid:0) R b S ) (cid:1) b x ] (cid:0) ( (cid:22) (cid:0) R )) (5.13)because at the optimal solution b x the constraint is binding. Using (r3) it follows from(5.2) and (5.12) that b r ( b x ) = (cid:21) E [( b S (cid:0) R b S ) (cid:1) b x ] = (cid:21) ( (cid:22) (cid:0) R ) = (cid:21) : (5.14)Thus, we can write (5.13) as b r ( b x ) (cid:21) b r ( b x ) E [( b S (cid:0) R b S ) (cid:1) b x ] : (5.15)For t (cid:21) x := (1 ; b ⊤ and x x t := ( tx + (1 (cid:0) t ) ; t b x ) : (5.16)We can verify that S (cid:1) x t = 1 and E [ S (cid:1) x t ] = R + t so that E [( S (cid:0) RS ) (cid:1) x t ] = t: (5.17)On the other hand it follows from assumptions (r1) and (r3) that r ( x t ) = b r ( t b x ) = t b r ( b x ) : (5.18)Thus, for any x satisfying S (cid:1) x = 1 and E [ S (cid:1) x ] (cid:21) R + t it follows from (5.15) that b r ( b x ) (cid:21) b r ( b x ) t: (5.19)30or any (cid:22) > R , letting t (cid:22) := (cid:22) (cid:0) R , we have (cid:22) = R + t (cid:22) . Thus, by inequality (5.19)we have b r ( b x ( (cid:22) )) (cid:21) t (cid:22) b r ( b x ). On the other hand x ( (cid:22) ) is an efficient portfolio implies that b r ( b x ( (cid:22) )) (cid:20) b r ( b x t (cid:22) ) = t (cid:22) b r ( b x ) yielding equality (cid:13) ( (cid:22) ) = b r ( b x ( (cid:22) )) = b r ( b x t (cid:22) ) = t (cid:22) b r ( b x ) = ( (cid:22) (cid:0) R ) b r ( b x ) : (5.20)In other words (cid:13) ( (cid:22) ) is an affine function in (cid:22) . Also, we conclude that points ( (cid:13) ( (cid:22) ) ; (cid:22) ) onthis efficient frontier correspond to efficient portfolios x t (cid:22) = ( ( (cid:22) (cid:0) R ) x + (cid:22) (cid:0) (cid:22); ( (cid:22) (cid:0) R ) b x ) = ( (cid:22) (cid:0) (cid:22) )(1 ; b ⊤ + ( (cid:22) (cid:0) R ) x (5.21)as an affine mapping of the parameter (cid:22) into the portfolio space.Also using r we can write (5.20) as (cid:13) ( (cid:22) ) = r ( (cid:22) (cid:0) R ) : (5.22)That is to say the efficient frontier of (5.1) in the risk-expected return space is given bythe parameterized straight line (5.6). Q.E.D. Remark 5.2. (a) Clearly, x t R corresponds to the portfolio (1 ; b ⊤ with (cid:13) ( R ) = b r ( b
0) = 0 .If x ̸ = 1 . Setting (cid:22) M := (cid:22) (cid:0) Rx (cid:0) x and r M := (cid:13) ( (cid:22) M ) = b r ( b x ) = (1 (cid:0) x ) we see that ( r M ; (cid:22) M ) on the efficient frontier corresponds to a purely risky efficient portfolio of (5.1) x M := x t (cid:22)M = ( ; (cid:0) x b x ) ⊤ : (5.23) Since x M belongs to the image of the affine mapping in (5.21), the family of efficientportfolios as described by the affine mapping in (5.21) contains both the pure bond (1 ; b ⊤ and the portfolio x M that consists only of purely risky assets. In fact, we can representthe affine mapping in (5.21) as a parametrized line passing through (1 ; b ⊤ and x M as x t (cid:22) = ( (cid:0) (cid:22) (cid:0) R(cid:22) M (cid:0) R ) (1 ; b ⊤ + (cid:22) (cid:0) R(cid:22) M (cid:0) R x M ; (5.24) which is a similar representation of the efficient portfolios as (5.5). The portfolio x M iscalled a master fund in [20]. When r = (cid:27) it is the market portfolio in the CAPM. For ageneral risk measure r satisfying conditions (r1), (r1n), (r2) and (r3) in Assumption 3.1the master funds x M are not necessarily unique. However, all master funds correspondto the same point ( r M ; (cid:22) M ) in the risk-expected return space.(b) We can also consider problem (5.1) on the set of admissible portfolios of purelyrisky assets, namely G eff ( r ; id; f S (cid:1) x = 1 ; x = 0 g ) . Then similar to the relationshipbetween the Markowitz efficient frontier and the capital market line, it follows from The-orem 5.1 that G eff ( r ; id; f S (cid:1) x = 1 ; x = 0 g ) \ G eff ( r ; id; f S (cid:1) x = 1 g ) = f ( r M ; (cid:22) M ) g ; (5.25)31 s illustrated in Figure 6.(c) If x = 1 then the efficient portfolios in (5.5) are related to (cid:22) in a much simplerfashion (1 ; b ⊤ + ( (cid:22) (cid:0) R ) b x : (5.26) In this case there is no master fund as observed in [20]. In the language of [20], portfolio x is called a basic fund . Thus, Theorem 5.1 recovers the results in Theorem 2 andTheorem 3 in [20] with a different proof and a weaker condition (condition (5.4) is weakerthan (A2) on page 752 of Rockafellar et al [20]). Since the standard deviation satis(cid:12)es Assumptions (r1), (r1n), (r2) and (r3), theresult above is a generalization of the relationship between the CAPM model and theMarkowitz portfolio theory. We note that the standard deviation is not the only riskmeasure that satis(cid:12)es these assumptions. For example, some forms of approximation tothe expected drawdowns also satisfy these assumptions (cf. [14]).Theorem 5.1 is a full generalization of the one fund theorem (Theorem 4.5) in theprevious section. On the other hand it has been noted in footnote 10 in [20] that asimilar generalization of the two fund theorem (Theorem 4.2) is not to be expected. Weconstruct a concrete counter-example below.
Example 5.3. (Counter-example to a Generalized Two Fund Theorem)
Let’s considerfor example min b x R b r ( b x ) (5.27)Subject to E [ b S (cid:1) b x ] (cid:21) (cid:22); b S (cid:1) b x = 1 ; with M = 3 .Choose all S m = 1 , so that b S (cid:1) b x = 1 is x + x + x = 1 . Choose the payoff S suchthat E [ b S (cid:1) b x ] = x so that x = (cid:22) at the optimal solution. Finally, let’s construct b r ( b x ) sothat the optimal solution b x ( (cid:22) ) is not affine in (cid:22) .We do so by constructing a convex set G with int G (interior of G ) and then set b r ( b x ) = 1 for b x @G (boundary of G ) and extend b r to be positive homogeneous. Then(r1), (r1n), (r2) and (r3) are satis(cid:12)ed.Now let’s specify G . Take the convex hull of the set [ (cid:0) ; (cid:2) [ (cid:0) ; (cid:2) [ (cid:0) ; and(cid:12)ve other points. One point is E = (10 ; ; ⊤ and the other four points A; B; C and D ,are the corner points of a square that lies in the plane x = 9 and has unit side length.To obtain that square take the standard square with unit side length in x = 9 , i.e. thesquare with corner points (9 ; (cid:6) = ; (cid:6) = ⊤ and rotate this square by 30 degrees counter (cid:22) ( r M ; (cid:22) M )(0 ; R ) Figure 6: Capital Market Line for (5.1) when x ̸ = 1 clockwise in the x x -plane. Doing some calculation one gets: A = (9 ; ( (cid:0) p = ; (1 + p = ⊤ B = (9 ; ( (cid:0) (cid:0) p = ; ( (cid:0) p = ⊤ C = (9 ; (1 (cid:0) p = ; (cid:0) (1 + p = ⊤ D = (9 ; (1 + p = ; (1 (cid:0) p = ⊤ : Obviously for (cid:22) = 1 the optimal solution is b x (1) = (1 ; ; ⊤ with b r ( b x (1)) = 1 = For (cid:22) = 1 + ϵ with ϵ > small we have b x (1 + ϵ ) = (1 + ϵ; ϵ p (cid:0) p = ; ϵ p (cid:0) (cid:0) p = ⊤ (they lie on the ray through a point on the convex combination of C and (10 ; ; ⊤ ) andfor (cid:22) = 1 + d with d > large we have b x (1 + d ) = (1 + d; (cid:0) d= ; (cid:0) d= ⊤ (they lie on theray through a point on the set f ( x ; (cid:0) ; (cid:0) ⊤ : x (2 ; g . Therefore, b x ( (cid:22) ) cannot beaffine in (cid:22) . Growth portfolio theory is proposed by Lintner [9] and is also related to the work of Kelly[8]. It is equivalent to maximizing the expected log utility:max x R M +1 E [ln( S (cid:1) x )] (6.1)Subject to S (cid:1) x = 1 : Remark 6.1.
Problem (6.1) is equivalent to max b x R M E [ln( R + ( b S (cid:0) R b S ) (cid:1) b x )] (6.2)33 heorem 6.2. (Growth Optimal Portfolio) Assume that the (cid:12)nancial market S t of Def-inition 2.1 has no nontrivial riskless portfolio. Then problem (6.1) has a unique opti-mal portfolio, which is often referred to as the growth optimal portfolio and is denoted (cid:20) R M +1 . To prove Theorem 6.2 we need the following lemma.
Lemma 6.3.
Assume that the (cid:12)nancial market S t of De(cid:12)nition 2.1 has no nontrivialriskless portfolio. Let u be a continuous utility function satisfying (u3) in Assumption3.3. Then for any (cid:22) R , f x R M +1 : E [ u ( S (cid:1) x )] (cid:21) (cid:22); S (cid:1) x = 1 g (6.3) is compact (and possibly empty in some cases). Proof.
Since u is continuous, the set in (6.3) is closed. Thus, we need only to show itis also bounded. Assume the contrary that there exists a sequence of portfolios x n with S (cid:1) x n = 1 (6.4)and ∥ x n ∥ ! 1 satisfying E [ u ( S (cid:1) x n )] (cid:21) (cid:22): (6.5)Equation (6.4) implies that ∥ b x n ∥ ! 1 . Then without loss of generality we may assume x n = ∥ b x n ∥ converges to x (cid:3) = ( x (cid:3) ; b x (cid:3) ) ⊤ where ∥ b x (cid:3) ∥ = 1. Condition (u3) and (6.5) forarbitrary (cid:22) R imply that, for each natural number n , S (cid:1) x n (cid:21) : (6.6)Dividing (6.4) and (6.6) by ∥ b x n ∥ and taking limits as n ! 1 we derive S (cid:1) x (cid:3) = 0 (6.7)and S (cid:1) x (cid:3) (cid:21) : (6.8)Combining (6.7) and (6.8) we have( b S (cid:0) R b S ) (cid:1) b x (cid:3) (cid:21) ; (6.9)and thus x (cid:3) is a nontrivial riskless portfolio, which is a contradiction. Q.E.D. Proof. of Theorem 6.2
We can verify that the utility function u = ln satis(cid:12)esconditions (u1), (u2s), (u3) and (u4). Also f x : E [ln( S (cid:1) x )] (cid:21) ln( R ) ; S (cid:1) x = 1 g ̸ = ∅ ; b ⊤ . Thus, Lemma 6.3 implies that problem (6.1) has at least onesolution and (cid:22) max = max x R M +1 f E [ln( S (cid:1) x )] : S (cid:1) x = 1 g is (cid:12)nite. By Lemma 3.14, x E [ln( S (cid:1) x )] is strictly concave. Thus problem (6.1) hasa unique optimal portfolio. Q.E.D.The growth optimal portfolio has the nice property that it provides the fastest com-pounded growth of the capital. By Remark 3.25 (b) it is independent of any risk measures.In the special case that all the risky assets are representing a certain gaming outcome, (cid:20) is the Kelly allocation in [8]. However, the growth portfolio is seldomly used in invest-ment practice for being too risky. The book [11] edited by MacLean, Thorp, and Ziembaprovides an excellent collection of papers with chronological research on this subject.These observations motivated Vince [28] to introduce his leverage space portfolio to scaleback from the growth optimal portfolio. Recently, [10, 30] further introduce systematicalmethods to scale back from the growth optimal portfolio by, among other ideas, explicitlyaccounts for limiting a certain risk measure. The analysis in [10, 30] can be phrased assolving (cid:13) ( (cid:22) ) := inf f r ( x ) = b r ( b x ) : E [ln( S (cid:1) x )] (cid:21) (cid:22); S (cid:1) x = 1 g ; (6.10)where r is a risk measure that satis(cid:12)es conditions (r1) and (r2). Alternatively, to derivethe efficient frontier we can also consider (cid:23) ( r ) := sup f E [ln( S (cid:1) x )] : r ( x ) = b r ( b x ) (cid:20) r; S (cid:1) x = 1 g ; (6.11)Applying Proposition 3.22, Theorem 3.24 and Remark 3.25 to the set of admissibleportfolios A = f x R M +1 : S (cid:1) x = 1 g we derive Theorem 6.4. (Leverage Space Portfolio and Risk Measure)
We assume that the (cid:12)-nancial market S t in De(cid:12)nition 2.1 has no nontrivial riskless portfolio and that the riskmeasure r satis(cid:12)es conditions (r1), (r1n) and (r2). Then(a) problem (6.10) de(cid:12)nes (cid:13) ( (cid:22) ) : [ln( R ) ; (cid:22) (cid:20) ] ! R as a continuous increasing convexfunction, where (cid:22) (cid:20) := E [ln( S (cid:1) (cid:20) )] and (cid:20) is the optimal growth portfolio. Moreover,problem (6.10) has a continuous path of unique solutions x ( (cid:22) ) that maps the interval [ln( R ) ; (cid:22) (cid:20) ] into a curve in the leverage portfolio space R M +1 . Finally, x (ln( R )) = (1 ; b ⊤ , x ( (cid:22) (cid:20) )) = (cid:20) , (cid:13) (ln( R )) = b r ( b
0) = 0 and (cid:13) ( (cid:22) (cid:20) ) = r ( (cid:20) ) .(b) problem (6.11) de(cid:12)nes (cid:23) ( r ) : [0 ; r ( (cid:20) )] ! R as a continuous increasing concavefunction, where (cid:20) is the optimal growth portfolio. Moreover, problem (6.11) has a con-tinuous path of unique solutions y ( r ) that maps the interval [0 ; r ( (cid:20) )] into a curve in theleverage portfolio space R M +1 . Finally, y (0) = (1 ; b ⊤ , y ( r ( (cid:20) )) = (cid:20) , (cid:23) (0) = ln( R ) and (cid:23) ( r ( (cid:20) )) = (cid:22) (cid:20) . Proof.
Note that Assumption 3.15 (a) holds due to Lemma 6.3 and (c2) in Theorem3.24 is also satis(cid:12)ed. Then (a) follows straight forward from conclusions (a) and (c) in35heorem 3.24 where (cid:22) max = (cid:22) (cid:20) and r min = 0 are (cid:12)nite and attained and (b) follows fromconclusions (b) and (c) in Theorem 3.24 with (cid:22) min = ln( R ) and r max = (cid:13) ( (cid:20) ). Q.E.D.Theorem 6.4 relates the leverage portfolio space theory to the framework setup inSection 3. It becomes clear that each risk measure satisfying conditions (r1), (r1n) and(r2) generates a path in the leverage portfolio space connecting the portfolio of a pureriskless bond to the growth optimal portfolio. Theorem 6.4 also tells us that different riskmeasures usually correspond to different paths in the portfolio space. Many commonlyused risk measures satisfy conditions (r1) and (r2). The curve x ( (cid:22) ) provides a pathwayto reduce risk exposure along the efficient frontier in the risk-expected log utility space.As observed in [10, 30], when investments have only a (cid:12)nite time horizon then there areadditional interesting points along the path x ( (cid:22) ) such as the in(cid:13)ection point and thepoint that maximizes the return/risk ratio. Both of which provide further landmarks forinvestors.Similar to the previous sections we can also consider the related problem of usingonly portfolios involving risky assets, i.e.,max b x R M E [ln( b S (cid:1) b x )] (6.12)Subject to b S (cid:1) b x = 1 : Theorem 6.5. (Existence of Solutions)
Suppose that S i ( ! ) > ; ! Ω ; i = 1 ; : : : ; M: (6.13) Then problem (6.12) has a solution.
Proof.
As in the proof of Theorem 6.4, we can see that Assumption 3.15 (a) holdsdue to Lemma 6.3. Observe that for b x (cid:3) = (1 =M; =M; : : : ; =M ) ⊤ we get from (6.13) that E [ln( b S (cid:1) b x (cid:3) )] is (cid:12)nite. Then we can directly apply Theorem 3.24 with A = f x R M +1 : S (cid:1) x = 1 ; x = 0 g . Q.E.D.However, due to the involvement of the log utility function, the relative location ofefficient frontiers (6.11) of (6.1) and (6.12) may have several different con(cid:12)gurations. Thefollowing is an example. Example 6.6.
Let M = 1 . Consider a sample space Ω = f ; g with probability P (0) =0 : and P (1) = 0 : and a (cid:12)nancial market involving a riskless bond with R = 1 andone risky asset speci(cid:12)ed by S = 1 , S (0) = 0 : and S (1) = 1 + (cid:11) with (cid:11) > = sothat E [ S ] > S . Use the risk measure r ( x ; x ) = j x j (which is an approximation ofthe drawdown cf. [30]). Then it is easy to calculate that the efficient frontier (6.11) of(6.1) is (cid:23) ( r ) = 0 :
55 ln(1 + (cid:11)r ) + 0 :
45 ln(1 (cid:0) : r ) ; r [0 ; r (cid:11) max ] ; (6.14) where r (cid:11) max = (22 (cid:11) (cid:0) = (cid:11) . On the other hand the efficient frontier of (6.12) is a singlepoint f (1 ; (cid:23) (1)) g where (cid:23) (1) = 0 :
55 ln(1 + (cid:11) ) (cid:0) :
45 ln(2) g . (cid:22) G ( r ; ln ; f S (cid:1) x = 1 g ) G ( r ; ln ; f S (cid:1) x = 1 ; x = 0 g ) Figure 7: Separated efficient frontiers
When (cid:11) (9 = ; = the two efficient frontiers (6.11) of (6.1) and (6.12) have nocommon points (see Figure 7). However, when (cid:11) (cid:21) = , G eff ( r ; ln ; S (cid:1) x = 1 ; x = 0) (cid:26)G eff ( r ; ln ; S (cid:1) x = 1) (see Figure 8). In particular, when (cid:11) = 9 = , G eff ( r ; ln ; S (cid:1) x =1 ; x = 0) coincide with the point on G eff ( r ; ln ; S (cid:1) x = 1) corresponding to the growthoptimal portfolio as illustrated in Figure 9.In fact, a far more common restriction to the set of admissible portfolios are limitsof risk. For this example if, for instance, we restrict the risk by r ( x ) (cid:20) : then we willcreate a shared efficient frontier of (6.1) with that of (6.11) where r is a priori restricted(see Figure 10). Remark 6.7. (Efficiency Index)
Although the growth optimal portfolio is usually notimplemented as an investment strategy, the maximum utility (cid:22) max corresponding to thegrowth optimal portfolio (cid:20) , empirically estimated using historical performance data, canbe used as a measure to compare different investment strategies. This is proposed in [31]and called the efficiency index. When the only risky asset is the payoff of a game withtwo outcomes following a given playing strategy, the efficiency coefficient coincides withShannon’s information rate (see [8, 21, 31]). In this sense, the efficiency index gaugesthe useful information contained in the investment strategy it measures.
Also related to the growth optimal portfolio theory is the fundamental theorem ofasset pricing (FTAP). FTAP characterizes the no arbitrage condition with the existenceof a martingale measure, which is de(cid:12)ned below.
De(cid:12)nition 6.8. (Equivalent Martingale Measure)
We say that Q is an equivalent mar-tingale measure (EMM) for the (cid:12)nancial market S t on a probability space (Ω ; Ω ; P ) provided that Q is a probability measure such that, for any ! Ω , Q ( ! ) ̸ = 0 if and onlyif P ( ! ) ̸ = 0 , and E Q [ S ] = RS : (cid:22) G ( r ; ln ; f S (cid:1) x = 1 g ) G ( r ; ln ; f S (cid:1) x = 1 ; x = 0 g ) Figure 8: Touching efficient frontiers r(cid:22) G ( r ; ln ; f S (cid:1) x = 1 g ) G ( r ; ln ; f S (cid:1) x = 1 ; x = 0 g ) Figure 9: Touching efficient frontiers at growth optimal38 (cid:22) G ( r ; ln ; f S (cid:1) x = 1 g ) G ( r ; ln ; f S (cid:1) x = 1 ; r ( x ) (cid:20) : g ) Figure 10: Shared efficient frontiersWe will relate the fundamental theorem of asset pricing to the following general utilityoptimization problem max x R M +1 E [ u ( S (cid:1) x )] (6.15)Subject to S (cid:1) x = 1 : First we observe that when a utility function u satis(cid:12)es condition (u4) we can alsocharacterize the no arbitrage condition in terms of the supremum of the expected utility. Theorem 6.9. (Characterization of No Arbitrage)
Suppose that the (cid:12)nancial market S t of De(cid:12)nition 2.1 has no nontrivial portfolio equivalent to the bond. Let u be a utilityfunction satisfying conditions (u3) and (u4) in Assumption 3.3. Then S t has no arbitrageif and only if sup x R M +1 f E [ u ( S (cid:1) x )] : S (cid:1) x = 1 g < + : Proof.
Note that f E [ u ( S (cid:1) x )] : S (cid:1) x = 1 g = f E [ u ( R + ( b S (cid:0) R b S ) (cid:1) b x )] : b x R M g : We can easily verify that when a utility function u satis(cid:12)es condition (u4) and thereexists an arbitrage portfolio thensup b x R M f E [ u ( R + ( b S (cid:0) R b S ) (cid:1) b x )] g = : On the other hand, by Proposition 3.7 when S t has no nontrivial portfolio equivalentto the bond and no arbitrage implies that S t has no nontrivial riskless portfolio. ByLemma 6.3, f x R M +1 : E [ u ( S (cid:1) x )] (cid:21) (cid:22); S (cid:1) x = 1 g is compact. Thus,sup x R M +1 f E [ u ( S (cid:1) x )] : S (cid:1) x = 1 g < + : Theorem 6.10. (Fundamental Theorem of Asset Pricing)
Suppose that the (cid:12)nancialmarket S t of De(cid:12)nition 2.1 has no nontrivial portfolio equivalent to the bond. Let u bea utility function that satis(cid:12)es properties (u1), (u2s), (u3) and (u4) in Assumption 3.3.Then the following assertions are equivalent: (i) The (cid:12)nancial market S t in De(cid:12)nition 2.1 has no arbitrage. (ii) The optimal value of the portfolio utility optimization problem (6.15) is (cid:12)nite andattained. (iii)
There is an equivalent martingale measure for the (cid:12)nancial market S t proportionalto a subgradient of (cid:0) u at the optimal solution of (6.15). Proof.
Observe that (i) equivalent to (ii) is already derived in Theorem 6.9.To prove (ii) implies (iii) we rewrite the utility optimization problem (6.15) asmax y E [ u ( y )] (6.16)subject to R + ( S ( ! ) (cid:0) RS ) (cid:1) x (cid:0) y ( ! ) = 0 ; for all ! Ω : Assume that ((cid:22) x; (cid:22) y ) is the solution to (6.16). Then there exist Lagrange multipliers (cid:21) ( ! ) P ( ! ), ! Ω such that the Lagrangian L (( x; y ) ; (cid:21) ) = E [ u ( y ) + (cid:21) ( R + ( S (cid:0) RS ) (cid:1) x (cid:0) y )] : (6.17)attains an unconstrained maximum at ((cid:22) x; (cid:22) y ). Thus, the convex function (cid:0) L attains anunconstrained minimum at ((cid:22) x; (cid:22) y ). It follows that (cid:0) (cid:21) ( ! ) @ ( (cid:0) u )((cid:22) y ( ! )) (6.18)(so that (cid:21) ( ! ) > E [ (cid:21) ( S (cid:0) RS )] = 0 : (6.19)It follows that Q = ( (cid:21)= E [ (cid:21) ]) P is an equivalent martingale measure. This process isreversible. Q.E.D. Following the pioneering idea of Markowitz to trade-off the expected return and standarddeviation of a portfolio, we consider a general framework to efficiently trade-off betweena concave expected utility and a convex risk measure for portfolios. Under reasonableassumptions we show that (i) the efficient frontier in such a trade-off is a convex curvein the expected utility-risk space, (ii) the optimal portfolio corresponding to each level40f the expected utility is unique and (iii) the optimal portfolios continuously depend onthe level of the expected utility. Moreover, we provide an alternative treatment of theresults in [20] showing that the one fund theorem (Theorem 4.5) holds in the trade-offbetween a deviation measure and the expected return (Theorem 5.1) and construct acounter-example illustrating that the two fund theorem (Theorem 4.2) fails in such ageneral setting. Furthermore, the efficiency curve in the leverage space is supposedly aneconomic way to scale back risk from the growth optimal portfolio (Theorem 6.4).This general framework uni(cid:12)es a group of well known portfolio theories. They areMarkowitz portfolio theory, capital asset pricing model, the growth optimal portfoliotheory, and the leverage portfolio theory. It also extends these portfolio theories to moregeneral settings.The new framework also leads to many questions of practical signi(cid:12)cance worthy fur-ther explorations. For example, quantities related to portfolio theories such as the Sharperatio and efficiency index can be used to measure investment performances. What otherperformance measurements can be derived using the general framework in Section 3?Portfolio theory can also inform us about pricing mechanisms such as those discussedin the capital asset pricing model and the fundamental theorem of asset pricing. Whatadditional pricing tools can be derived from our general framework?Clearly, for the purpose of applications we need to focus on certain special cases.Drawdown related risk measures coupled with the log utility attracts much attentionin practice. In Part II of this series [14] several drawdown related risk measures areconstructed and analyzed. We will conduct a related case study in the third part of thisseries [3].
References [1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk.
Mathe-matical Finance , 9:203-227, 1999.[2] D. Bernoulli, Exposition of a new theory on the measurement of risk.
Econometrica ,22:23{36, 1954/1738.[3] R. Brenner, S. Maier-Paape, A. Platen and Q. J. Zhu, A general framework forportfolio theory. Part III: applications. In preparation.[4] J.M. Borwein and Q.J. Zhu,
Techniques of Variational Analysis . Springer-Verlag,2005.[5] J.M. Borwein and Q.J. Zhu, A variational approach to Lagrange multipliers.
J.Optimization Theory and Applications , 171:727-756, 2016.[6] P. Carr and Q. J. Zhu,
Convex Duality and Financial Mathematics . Springer-Verlag,to appear. 417] W. Fenchel,
Convex Cones, Sets and Functions . Lecture Notes, Princeton University,Princeton, 1951.[8] J. L. Kelly, A new interpretation of information rate.
Bell System Technical Journal ,35:917{926, 1956.[9] J. Lintner, The valuation of risk assets and the selection of risky investments instock portfolios and capital budgets.
Review of Economics and Statistics , 47:13{37,1965.[10] M. Lopez de Prado, R. Vince, and Q. J. Zhu, Optimal risk budgeting under a (cid:12)niteinvestment horizon. SSRN 2364092, 2013.[11] L. C. MacLean, E. O. Thorp, and W. T. Ziemba(Eds.).
The Kelly Capital GrowthCriterion: Theory and Practice . World Scienti(cid:12)c, 2009.[12] S. Maier-Paape, Optimal f and diversi(cid:12)cation. IFTA Journal , 4{7, 2015.[13] S. Maier-Paape, Risk averse fractional trading using the current drawdown.
Institutf(cid:127)ur Mathematik , RWTH Aachen, Report no. 88, 2016.[14] S. Maier-Paape and Q. J. Zhu, A general framework for portfolio theory. Part II:drawdown risk measures.
Institut f(cid:127)ur Mathematik , RWTH Aachen, Report no. 92,2017.[15] H. Markowitz,
Portfolio Selection . Cowles Monograph, 16. Wiley, New York, 1959.[16] J. J. Moreau,
Fonctionelles Convexes . College de France, Lecture notes, 1967.[17] J. Mossin, Equilibrium in a capital asset market.
Econometrica , 34:768 {783, 1966.[18] R. T. Rockafellar,
Convex Analysis . Vol. 28 Princeton Math. Series, Princeton Uni-versity Press, Princeton, 1970.[19] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk.
Journalof Risk
J. Banking and Finance
The Mathematical Theory of Communication . Urbana,Illinois: University of Illinois Press, 1949.[22] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditionsof risk.
Journal of Finance , 19:425{442, 1964.[23] W. F. Sharpe, Mutual fund performance.
Journal of Business , 1:119{138, 1966.4224] E. O. Thorp and S. T. Kassouf,
Beat the Market . Random House, New York, 1967.[25] J. L. Treynor, Toward a theory of market value of risky assets. Unpublishedmanuscript 1962. A (cid:12)nal version was published in 1999, in
Asset Pricing and Portfo-lio Performance: Models, Strategy and Performance Metrics . Robert A. Korajczyk(editor) London: Risk Books, pp. 1522.[26] J. Tobin, Liquidity preference as behavior towards risk.
The Review of EconomicStudies , 26:65-86, 1958.[27] R. Vince,
The New Money Management: A Framework for Asset Allocation . JohnWiley and Sons, New York, 1995.[28] R. Vince,
The Leverage Space Trading Model . John Wiley and Sons, Hoboken, NJ,2009.[29] R. Vince and Q. J. Zhu, In(cid:13)ection point signi(cid:12)cance for the investment size.
SSRN ,2230874, 2013.[30] R. Vince and Q. J. Zhu, Optimal betting sizes for the game of blackjack.
RiskJournals: Portfolio Management , 4:53-75, 2015.[31] Q. J. Zhu, Mathematical analysis of investment systems.
Journal of MathematicalAnalysis and Applications , 326:708-720, 2007.[32] Q. J. Zhu, Convex analysis in mathematical (cid:12)nance.